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Solving Exponential Equations using Logarithms

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Solving Exponential Equations using Logarithms

Study Guide

Key Definition

Logarithms are the inverse of exponential functions. If $a^x = b$, then $x = \log_a(b)$.

Important Notes

Exponential equations can be solved by taking logarithms on both sides.
Ensure to isolate the exponential expression before taking the logarithm.
Use a calculator for more precise solutions after taking logarithms.
Logarithmic expressions can be rewritten in exponential form.
Approximate solutions may involve the ≈ symbol.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\log_a(b)$ means logarithm of b with base a

Remember to use proper notation when solving problems

Why It Works

Logarithms undo exponents because they are inverse operations. This allows us to solve for variables in exponential equations by converting them into logarithmic form.

Remember

Always check your work by substituting your solution back into the original equation. Use $\log$ properties to simplify expressions.

Quick Reference

Exponential Form:$a^x = b$

Logarithmic Form:$x = \log_a(b)$

Change of Base Formula:$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$

Understanding Solving Exponential Equations using Logarithms

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Basic concept: to solve $a^x = b$, isolate the exponential term and apply logarithm to both sides. Then use the inverse relationship $x = \log_a(b)$ to find x.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $2^x = 8$, what is $x$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A new tech gadget's price drops by half each year. If its original price was $1000$, when will it be approximately $250$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Solve for x in $5^{2x} = 125$.

Challenge Quiz

Single Choice Quiz

Advanced

Solve $3^{x+1} = 81$.

Recap

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Review key concepts and takeaways