Question

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Rewrite the logarithm as a ratio of common logarithms and natural logarithms.
log_1/3(4)
a. common logarithms

b. natural logarithms

Answer 1

Common logarithm is the logarithm with the base 10, it is written as:

• log a, lg a or log₁₀ a

Natural logarithm is the logarithm with the base of e and is written as ln a.

How to rewrite a logarithm $log_a\ b$ as a ratio of logarithms with a different base c:

• $log_a\ b=log_c\ b\ /\ log_c\ a$

Apply this to the given logarithm:

a. Common logarithms:

• $log_{1/3}\ 4=log\ 4\ /\ log\ (1/3)$

b. Natural logarithms:

• $log_{1/3}\ 4=ln\ 4\ /\ ln\ (1/3)$

Answer 2

Answer:

$\textsf{a)} \quad \dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}=-\dfrac{\log_{10}4}{\log_{10}3}$

$\textsf{b)} \quad \dfrac{\ln 4}{\ln \dfrac{1}{3}}=-\dfrac{\ln 4}{\ln 3}$

Step-by-step explanation:

Given logarithm:

$\log_{\frac{1}{3}}(4)$

Part (a)

The common logarithm is the logarithm with base 10.

$\boxed{\textsf{Change of base}: \quad \log_ba=\dfrac{\log_xa}{\log_xb}}$

Use the change of base formula to rewrite the given logarithm as a ratio of common logarithms:

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}$

$\boxed{\begin{minipage}{5cm} \underline{Log quotient law}\\\\ \log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay \\\end{minipage}}$

This can be simplified using the log quotient rule:

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}$

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}1-\log_{10}3}$

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{0-\log_{10}3}$

$\implies \log_{\frac{1}{3}}(4)=-\dfrac{\log_{10}4}{\log_{10}3}$

Part (b)

The natural logarithm is the logarithm with base e.

$\textsf{Also} \; \log_ex=\ln x$

Use the change of base formula to rewrite the given logarithm as a ratio of natural logarithms:

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{e}4}{\log_{e}\dfrac{1}{3}}$

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln \dfrac{1}{3}}$

$\boxed{\begin{minipage}{5cm} \underline{Log quotient law}\\\\ \ln \left(\dfrac{x}{y}\right)=\ln x - \ln y \\\end{minipage}}$

This can be simplified using the log quotient rule:

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln \dfrac{1}{3}}$

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln1 - \ln 3}$

$\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{0 - \ln 3}$

$\implies \log_{\frac{1}{3}}(4)=-\dfrac{\ln 4}{\ln 3}$