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Chapter 13: Problem 33

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log \sqrt[3]{100}$$

Short Answer

Expert verified
\(\log \sqrt[3]{100}=\frac{2}{3}\)

Step by step solution

01

Rewrite the cube root as a power

Rewrite the cube root of \(100\) as \(100\) raised to the power of \(\frac{1}{3}\). The expression can be rewritten as: $$\log \left(100^{\frac{1}{3}}\right)$$
02

Apply the logarithm power rule

Use the power rule of logarithms, which states that \(\log (a^{b}) = b\log (a)\), to rewrite the expression: $$\frac{1}{3}\log (100)$$
03

Simplify the logarithm

Since we have the logarithm of \(100\), we can simplify the expression as follows: $$ \frac{1}{3}\log (100) = \frac{1}{3}\log (10^2) $$ Now, apply the power rule again by bringing down the exponent: $$ \frac{1}{3}(2\log (10)) $$ Remember that the logarithm of \(10\) to the base \(10\) is equal to \(1\): $$ \frac{1}{3}(2(1)) $$
04

Evaluate the expression

Evaluate the expression: $$ \frac{1}{3}(2) = \frac{2}{3} $$ Therefore, the expression can be rewritten and simplified as: $$\log \sqrt[3]{100}=\frac{2}{3}$$

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Most popular questions from this chapter

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

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 \(1 .\) $$\frac{1}{3} \log _{a} 5-2 \log _{a} z$$

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The hydronium ion concentrations, \(\left[\mathrm{H}^{+}\right],\) are given for some common substances. Find the \(\mathrm{pH}\) of each substance (to the tenths place), and determine whether each substance is acidic or basic. $$\text { Egg white: }\left[\mathrm{H}^{+}\right]=2 \times 10^{-8}$$
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