Solving Logarithms - Algebra 2

To solve for , first convert both sides to the same base:

Now, with the same base, the exponents can be set equal to each other:

Solving for gives:

Rewrite in exponential form:

Solve for x:

Logs are exponential functions using base 10 and a property is that you can combine added logs by multiplying.

You cannot take the log of a negative number. x=-25 is extraneous.

Rewrite as a product that includes the number :

Then we can split up the logarithm using the Product Property of Logarithms:

Thus,

.

To solve this logarithm, we need to know how to read a logarithm. A logarithm is the inverse of an exponential function. If a exponential equation is

then its inverse function, or logarithm, is

Therefore, for this problem, in order to solve for , we simply need to solve

which is .

For this problem it is helpful to remember that,

is equivalent to because

Therefore we can set what is inside of the parentheses equal to each other and solve for as follows:

This question is testing the definition of logs. is the same as .

In this case, can be rewritten as .

Taking square roots of both sides, you get . Since only the positive answer is one of the answer choices, is the correct answer.

The first step is to rewrite this equation in log form.

When rewriting an exponential function as a log we must remember that the form of an exponential is:

When this is rewritten in log form it is:

.

Therefore we have which when rewritten gives us,

.

Logarithms are another way of writing exponents. In the general case, really just means . We take the base of the logarithm (in our case, 2), raise it to whatever is on the other side of the equal sign (in our case, 4) and set that equal to what is inside the parentheses of the logarithm (in our case, x+6). Translating, we convert our original logarithm equation into . The left side of the equation yields 16, thus .

To solve this equation, remember log rules

.

This rule can be applied here so that

and

Because both sides have the same logarithmic base, both terms can be set equal to each other:

Now, evaluate the equation.

First, add x to both sides:

Add 15 to both sides:

Finally, divide by 6: .

In logarithmic expressions, is the same thing as .

Therefore, the equation can be rewritten as .

Both 8 and 128 are powers of 2, so the equation can then be rewritten as .

Since both sides have the same base, set .

Solve by dividing both sides of the equation by 3: .

Use the rule of Exponents of Logarithms to turn all the multipliers into exponents:

.

Simplify by applying the exponents: .

According to the law for adding logarithms, .

Therefore, multiply the 4 and 7.

.

Because both sides have the same base, .

To solve this problem you must be familiar with the one-to-one logarithmic property.

if and only if x=y. This allows us to eliminate to logarithmic functions assuming they have the same base.

one-to-one property:

isolate x's to one side:

move constant:

Get all the terms with e on one side of the equation and constants on the other.

Apply the logarithmic function to both sides of the equation.

Recall the rules of logs to solve this problem.

First, when there is a coefficient in front of log, this is the same as log with the inside term raised to the outside coefficient.

Also, when logs of the same base are added together, that is the same as the two inside terms multiplied together.

In mathematical terms:

Thus our equation becomes,

To simplify further use the rule,

.

To solve this equation, you must first simplify the log expressions and remember log laws (; ). Therefore, for the first expression, so . For the second expression, so Then, substitute those numbers in so that you get .

Logarithms are inverses of exponents.

is asking how many fives multiplied together are equal to 25.

The formula to solve logarithmic equations is as follows.

Applying this formula to our particular problem results in the following.

=

So .

Remember that a logarithm is nothing more than an exponent. The equation reads, the logarithm (exponent) with base that gives is .