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Chapter 3: Problem 15

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log _{4} 8 $$

Short Answer

Expert verified
The value of the given expression \(\log_4 8\) is \(\frac{3}{2}\).

Step by step solution

01

Rewrite the given expression as an exponential equation

We can rewrite the given expression, \(\log_4 8\), as an exponential equation as follows: $$ 4^x = 8 $$ Now, we need to solve for x which is the value of the given logarithm expression.
02

Convert the base 4 logarithm to base 2

The given expression has base 4, which is 2 raised to the power of 2 (since \(4=2^2\)). To solve for x, we can convert the base 4 logarithm to base 2. To do this, we use the following property of logarithms, if \(a = b^c\), then \(\log_b(a) = c\). Therefore, we can write the expression as: $$ (2^2)^x = 8 $$
03

Rewrite the expression with base 2

Now, we can rewrite the expression using base 2 logarithms by applying the power rule for exponents: \(a^{bc} = (a^b)^c\). Thus, we have: $$ 2^{2x} = 8 $$ Now, we need to rewrite the number 8 in terms of base 2 exponentials (i.e., 8 as a power of 2).
04

Rewrite the number 8 as a power of 2

Since \(2^3 = 8\), we can rewrite the expression as: $$ 2^{2x} = 2^3 $$
05

Solve for x

Now that we have the expression in terms of the same base 2 exponentials, we can equate the exponents and solve for x: $$ 2x = 3 $$ Divide both sides of the equation by 2 to get x: $$ x=\frac{3}{2} $$ Now we have evaluated the logarithm expression without a calculator. Thus, the value of the given expression \(\log_4 8\) is \(\frac{3}{2}\).

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