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

`\textsf{a)} \quad (\log_(10)4)/(\log_(10)(1)/(3))=-(\log_(10)4)/(\log_(10)3)`

`\textsf{b)} \quad (\ln 4)/(\ln (1)/(3))=-(\ln 4)/(\ln 3)`

Explanation:

Given logarithm:

`\log_{(1)/(3)}(4)`

Part (a)

The common logarithm is the logarithm with base 10.

`\boxed{\textsf{Change of base}: \quad \log_ba=(\log_xa)/(\log_xb)}`

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

`\implies \log_{(1)/(3)}(4)=(\log_(10)4)/(\log_(10)(1)/(3))`

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

This can be simplified using the log quotient rule:

`\implies \log_{(1)/(3)}(4)=(\log_(10)4)/(\log_(10)(1)/(3))`

`\implies \log_{(1)/(3)}(4)=(\log_(10)4)/(\log_(10)1-\log_(10)3)`

`\implies \log_{(1)/(3)}(4)=(\log_(10)4)/(0-\log_(10)3)`

`\implies \log_{(1)/(3)}(4)=-(\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_{(1)/(3)}(4)=(\log_(e)4)/(\log_(e)(1)/(3))`

`\implies \log_{(1)/(3)}(4)=(\ln 4)/(\ln (1)/(3))`

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

This can be simplified using the log quotient rule:

`\implies \log_{(1)/(3)}(4)=(\ln 4)/(\ln (1)/(3))`

`\implies \log_{(1)/(3)}(4)=(\ln 4)/(\ln1 - \ln 3)`

`\implies \log_{(1)/(3)}(4)=(\ln 4)/(0 - \ln 3)`

`\implies \log_{(1)/(3)}(4)=-(\ln 4)/(\ln 3)`

Answer 2

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)`