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Chapter 3: Problem 16

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log _{8} 128 $$

Short Answer

Expert verified
The short answer to the problem \(\log_8 128\) is 2. This is because \(8^2 = 128\).

Step by step solution

01

Rewrite the log expression

We have the logarithm expression \(\log_{8} 128\). We need to rewrite it as an exponential equation with base 8. The exponential equation will look like this: \(8^x = 128\).
02

Rewrite 128 as a power of 8

We need to rewrite the number 128 as a power of 8 (with the same base as the logarithm expression). 128 can be rewritten as \(8^2 \times 2 = (2^3)^2 \times 2 = 2^{3\times2}\), which is equal to \(2^6\).
03

Express both sides of the equation with the same base

We have the equation \(8^x = 2^6\). The bases of both values are not the same. Since 8 can also be represented as a power of 2, express the equation in terms of base 2. We know that \(8=2^3\), so the equation becomes \((2^3)^x = 2^6\).
04

Solve the equation for x

Now that we have an equation with the same base, \((2^3)^x = 2^6\), we can use the properties of exponential functions to solve for x. Using the exponentiation rule for powers, we can multiply the exponents: \(2^{3x} = 2^6\). Since the bases are equal, the exponents must also be equal: \(3x = 6\). Solve for x by dividing both sides by 3: \(x = \frac{6}{3} = 2\).
05

Write the value of the logarithm expression

We found that the value of x is 2. Substitute this value back into the original logarithm expression, \(\log_{8} 128\). Thus, the value of the expression is 2: \(\log_{8} 128 = 2\).

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Most popular questions from this chapter

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\log _{5 x} 6 $$

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