This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like $2^t$ = 10 where the exponent contains the variable, logarithms are the tool that unlocks the solution: taking log base 2 of both sides gives $log₂(2^t$) = log₂(10), and using the inverse property $log₂(2^t$) = t, we get t = log₂(10). This is the exact solution! To get a decimal approximation, use your calculator with change of base: log₂(10) = ln(10)/ln(2) ≈ 3.322. To solve $4·2^{t/5}$=30, divide by 4: $2^{t/5}$=30/4=15/2, take log base 2: $log₂(2^{t/5}$)=log₂(15/2), simplify to t/5=log₂(15/2), and multiply by 5: t=5 log₂(15/2). Choice A correctly isolates, takes log base 2, applies the inverse property, and approximates to 14.535 using change of base on a calculator. Choice B divides by 5 instead of multiplying, resulting in t=log₂(15/2)/5 which is one-fifth the size—remember that to undo division in the exponent, you multiply when solving for t! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for 8·e^(t/3) = 90: (1) Isolate the exponential by dividing both sides by 8: e^(t/3) = 90/8 = 45/4, (2) Take ln of both sides: ln(e^(t/3)) = ln(45/4), (3) Use inverse property on left: t/3 = ln(45/4), (4) Solve for variable: t = 3ln(45/4). This systematic approach works for any exponential equation in this form! Starting with 8·e^(t/3) = 90, divide by 8 to get e^(t/3) = 45/4, then take ln of both sides to get t/3 = ln(45/4), and multiply by 3 to get t = 3ln(45/4). Using a calculator: ln(45/4) = ln(11.25) ≈ 2.422, so t ≈ 3(2.422) ≈ 7.266. Choice A correctly isolates the exponential, takes the natural logarithm, and multiplies by 3 to get t = 3ln(45/4) ≈ 7.266. Choice B incorrectly divides by 3 instead of multiplying—when the exponent is t/3 and you get t/3 = ln(45/4), you must multiply both sides by 3 to solve for t! Isolation before logarithms: ALWAYS isolate the exponential expression e^(t/3) before taking logarithms. If you have 8e^(t/3) = 90, first divide by 8 to get e^(t/3) = 45/4, THEN take ln. Simple isolation first makes the logarithm application clean!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like 15·2^(2t) = 500 where the exponent contains the variable, logarithms are the tool that unlocks the solution: first isolate the exponential by dividing both sides by 15 to get 2^(2t) = 500/15 = 100/3, then take log base 2 of both sides to get log₂(2^(2t)) = log₂(100/3), and using the inverse property log₂(2^(2t)) = 2t, we get 2t = log₂(100/3), so t = log₂(100/3)/2. Starting with 15·2^(2t) = 500, divide by 15 to get 2^(2t) = 100/3, then take log₂ of both sides: 2t = log₂(100/3), and divide by 2 to get t = log₂(100/3)/2. To approximate with a calculator, use change of base: log₂(100/3) = ln(100/3)/ln(2) ≈ 3.507/0.693 ≈ 5.058, so t ≈ 5.058/2 ≈ 2.529. Choice A correctly isolates the exponential, takes log base 2, and divides by 2 to get t = log₂(100/3)/2 ≈ 2.529. Choice B incorrectly keeps the original fraction 500/15 without simplifying—while 500/15 = 100/3, the simplified form makes the calculation clearer! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(100/3)? Use change of base: log₂(100/3) = ln(100/3)/ln(2) ≈ 5.058. The formula is log_b(x) = ln(x)/ln(b) for any base b.

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like 9·e^(2t) = 140 where the exponent contains the variable, logarithms are the tool that unlocks the solution: first isolate the exponential by dividing both sides by 9 to get e^(2t) = 140/9, then take ln of both sides to get ln(e^(2t)) = ln(140/9), and using the inverse property ln(e^(2t)) = 2t, we get 2t = ln(140/9), so t = ln(140/9)/2. Starting with 9·e^(2t) = 140, divide by 9 to get e^(2t) = 140/9, then take ln of both sides: 2t = ln(140/9), and divide by 2 to get t = ln(140/9)/2. Using a calculator: ln(140/9) ≈ ln(15.556) ≈ 2.747, so t ≈ 2.747/2 ≈ 1.374. Choice A correctly isolates the exponential, takes the natural logarithm, and divides by 2 to get t = ln(140/9)/2 ≈ 1.374. Choice C incorrectly multiplies by 2 instead of dividing—remember that when you have 2t = ln(140/9), you must divide both sides by 2 to solve for t! Isolation before logarithms: ALWAYS isolate the exponential expression e^(2t) before taking logarithms. If you have 9e^(2t) = 140, first divide by 9 to get e^(2t) = 140/9, THEN take ln. This makes the logarithm application clean: take ln of both sides when you have e^(something) = number!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for 5·2^(3t) = 60: (1) Isolate the exponential by dividing both sides by 5: 2^(3t) = 12, (2) Take log base 2 of both sides: log₂(2^(3t)) = log₂(12), (3) Use inverse property on left: 3t = log₂(12), (4) Solve for variable: t = log₂(12)/3. This systematic approach works for any exponential equation in this form! Starting with 5·2^(3t) = 60, divide by 5 to get 2^(3t) = 12, then take log₂ of both sides to get 3t = log₂(12), and finally divide by 3 to get t = log₂(12)/3. To approximate with a calculator, use change of base: log₂(12) = ln(12)/ln(2) ≈ 3.585, so t ≈ 3.585/3 ≈ 1.195. Choice A correctly isolates the exponential, takes log base 2, and divides by 3 to get t = log₂(12)/3 ≈ 1.195. Choice B incorrectly uses 60 instead of 12 after isolation—remember to divide 60 by 5 first to get 2^(3t) = 12, not 2^(3t) = 60! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(12)? Use change of base: log₂(12) = ln(12)/ln(2) or log(12)/log(2)—both give the same answer ≈ 3.585. The formula is log_b(x) = ln(x)/ln(b) for any base b.

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for ab^(ct) = d: (1) Isolate the exponential by dividing both sides by a: b^(ct) = d/a, (2) Take log base b of both sides: log_b(b^(ct)) = log_b(d/a), (3) Use inverse property on left: ct = log_b(d/a), (4) Solve for variable: t = log_b(d/a)/c. This systematic approach works for any exponential equation in this form! To solve $3·10^{2x}$=75, first divide both sides by 3 to isolate the exponential: $10^{2x}$=25, then take the common logarithm (base 10) of both sides: $log(10^{2x}$)=log(25), which simplifies to 2x=log(25) using the inverse property, and finally divide by 2: x=log(25)/2. Choice A correctly isolates and takes the logarithm to get the exact solution as log(25)/2 and approximates it to 0.699 using technology. Choice C forgets to divide by 2 after taking the log, resulting in x=log(25) which is twice as large as it should be—remember to solve fully for the variable after applying the logarithm! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for 12·2^(t/4) = 100: (1) Isolate the exponential by dividing both sides by 12: 2^(t/4) = 100/12 = 25/3, (2) Take log base 2 of both sides: log₂(2^(t/4)) = log₂(25/3), (3) Use inverse property on left: t/4 = log₂(25/3), (4) Solve for variable: t = 4log₂(25/3). This systematic approach works for any exponential equation in this form! Starting with 12·2^(t/4) = 100, divide by 12 to get 2^(t/4) = 25/3, then take log₂ of both sides to get t/4 = log₂(25/3), and multiply by 4 to get t = 4log₂(25/3). To approximate with a calculator, use change of base: log₂(25/3) = ln(25/3)/ln(2) ≈ 2.120/0.693 ≈ 3.059, so t ≈ 4(3.059) ≈ 12.235. Choice A correctly isolates the exponential, takes log base 2, and multiplies by 4 to get t = 4log₂(25/3) ≈ 12.235. Choice B incorrectly divides by 4 instead of multiplying—when the exponent is t/4 and you get t/4 = log₂(25/3), you must multiply both sides by 4 to solve for t! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(25/3)? Use change of base: log₂(25/3) = ln(25/3)/ln(2) ≈ 3.059. Always multiply by the coefficient when the variable appears as t/4 in the exponent!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for 5·2^(3x) = 60: (1) Isolate the exponential by dividing both sides by 5: 2^(3x) = 12, (2) Take log base 2 of both sides: log₂(2^(3x)) = log₂(12), (3) Use inverse property on left: 3x = log₂(12), (4) Solve for variable: x = log₂(12)/3. This systematic approach works for any exponential equation in this form! Starting with 5·2^(3x) = 60, divide by 5 to get 2^(3x) = 12, then take log₂ of both sides: log₂(2^(3x)) = log₂(12), which simplifies to 3x = log₂(12), so x = log₂(12)/3. To evaluate with a calculator, use change of base: log₂(12) = ln(12)/ln(2) ≈ 3.585/1.386 ≈ 3.585, so x ≈ 3.585/3 ≈ 1.195. Choice A correctly isolates the exponential, takes log base 2, and divides by 3 to get x = log₂(12)/3 ≈ 1.195. Choice B incorrectly uses log₂(60) without first isolating the exponential, while Choice D uses common log instead of log base 2 as requested. Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(12)? Use change of base: log₂(12) = ln(12)/ln(2) or log(12)/log(2)—both give the same answer ≈ 3.585. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like $2^t$ = 10 where the exponent contains the variable, logarithms are the tool that unlocks the solution: taking log base 2 of both sides gives $log₂(2^t$) = log₂(10), and using the inverse property $log₂(2^t$) = t, we get t = log₂(10). This is the exact solution! To get a decimal approximation, use your calculator with change of base: log₂(10) = ln(10)/ln(2) ≈ 3.322. To solve $5·2^{3t}$=60, first divide both sides by 5 to isolate: $2^{3t}$=12, then take log base 2: $log₂(2^{3t}$)=log₂(12), which simplifies to 3t=log₂(12), and divide by 3: t=log₂(12)/3. Choice A correctly isolates the exponential, applies the logarithm base 2, uses the inverse property, and approximates to 1.195. Choice C omits dividing by 3, leaving t=log₂(12) which is three times larger than needed—always remember to divide by the coefficient in the exponent after taking the log! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like 9·2^(x/5) = 40.5 where the exponent contains the variable, logarithms are the tool that unlocks the solution: first isolate the exponential by dividing both sides by 9 to get 2^(x/5) = 4.5, then take log₂ of both sides to get log₂(2^(x/5)) = log₂(4.5), and using the inverse property log₂(2^(x/5)) = x/5, we get x/5 = log₂(4.5), so x = 5·log₂(4.5). To get a decimal approximation, use change of base: log₂(4.5) = ln(4.5)/ln(2) ≈ 1.504/0.693 ≈ 2.170, so x ≈ 5(2.170) ≈ 10.85. Choice A correctly isolates the exponential term, takes log base 2, and multiplies by 5 to get x = 5·log₂(4.5) ≈ 10.85. Choice B incorrectly divides by 5 instead of multiplying, while Choice C uses log₂(40.5) without first isolating the exponential, and Choice D uses common log instead of log base 2 as requested. Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(4.5)? Use change of base: log₂(4.5) = ln(4.5)/ln(2) or log(4.5)/log(2)—both give the same answer ≈ 2.170. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button!