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Chapter 13: Problem 46

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{8} \sqrt[3]{\frac{z}{8}}$$

Short Answer

Expert verified
The simplified expression is \(\frac{1}{3}\left(\log_8{(z)}-1\right)\).

Step by step solution

01

Rewrite the cube root as a power of one-third

The first step is to rewrite the cube root as a power of one-third. This can be done as follows: $$\log_8 \left(\frac{z}{8}\right)^{\frac{1}{3}}$$ Now we have converted the cube root, and we are ready to apply the logarithm properties.
02

Apply the logarithm power rule

Using the logarithm power rule, we can distribute the exponent to the logarithm: $$\frac{1}{3}\log_8{\frac{z}{8}}$$ Now let's use the quotient rule for logarithms to simplify the expression further.
03

Apply the quotient rule for logarithms

The quotient rule for logarithms states that \(\log_b(a/c)=\log_b(a)-\log_b(c)\). Applying this to our expression, we get: $$\frac{1}{3}\left(\log_8{(z)}-\log_8{(8)}\right)$$
04

Simplify the expression

Now, let's simplify the second term of the expression. We know that \(\log_8{8}=1\), so our final expression becomes: $$\frac{1}{3}\left(\log_8{(z)}-1\right)$$ Hence, the given logarithm expression is simplified as the sum or difference of logarithms as: $$\log _{8} \sqrt[3]{\frac{z}{8}} = \frac{1}{3}\left(\log_8{(z)}-1\right)$$

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Most popular questions from this chapter

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$4 \log _{3} f+\log _{3} g$$

Given that \(\log 5=0.6990\) and \(\log 9=0.9542,\) use the propertics of logarithms to approximate the following $$\log \frac{1}{9}$$

Find the inverse of each one-to-one function. $$g(x)=\sqrt[3]{x+2}$$

Given that \(\log 5=0.6990\) and \(\log 9=0.9542,\) use the propertics of logarithms to approximate the following $$\log \frac{25}{9}$$

Given that \(\log 5=0.6990\) and \(\log 9=0.9542,\) use the propertics of logarithms to approximate the following $$\log 90$$
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