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# Finding Logarithms

You can find the value of some logarithms by first writing them in exponential form. Once the equation is in exponential form, you can solve it using the Principle of Exponential Equality.

Note:

Recall that the Principle of Exponential Equality states that if bx = by, then x = y.

Here, b > 0 and b 1.

Example 1

Find log232.

Solution
 Let x equal log232 to create an equation. Rewrite in exponential form. Rewrite 32 as 25. Use the Principle of Exponential Equality. log232 = x 2x = 32 2x = 25 x = 5

So, log232 = 5.

Here is a check:

Is log232 = 5 ?

Is 25 = 32 ?

Is 32 = 32 ? Yes

Example 2

Find log51.

Solution
 Let x equal log51 to create an equation. Rewrite in exponential form.  Rewrite 1 as 50. Use the Principle of Exponential Equality. log51 = x 5x = 1 5x = 50 x = 0

Thus, log51 = 0.

Note:

Sometimes one side of an exponential equation cannot be easily rewritten as a power with the same base as the other side. Later, you will learn how to solve such equations.

Example 3

Find Solution
 Let x equal to create an equation. Rewrite in exponential form. Rewrite Write the right side as 3-2. Use the Principle of Exponential Equality. = x  3x = 3-2 x = -2

So, Note:

Recall the defination of a negative exponent: Example 4

Find log 1/2 8.

Solution

 Let x equal log 1/2 8 to create an equation. Rewrite in exponential form. Rewrite as 2-x and 8 as 23. Use the Principle of Exponential Equality. Multiple or divide both sides by -1. log 1/2 8 = x = 8 2-x = 23 -x = 3 x = -3

Therefore, log 1/2 8 = -3.

Note: 