1 Answer

Ask a Question

Ajit Panigrahi · Answer 1 · 2023-10-15T20:48:20+0000

Hello there. To solve this question, we'll have to remember some properties about changing bases in logarithms.

Given the logarithm:

$\log_3\left((1)/(4)\right)$

We first use the properties:

$\begin{gathered} \log_c\left((a)/(b)\right)=\log_c(a)-\log_c(b)\text{ and } \\ \\ \log_c(1)=0,0In this case, we get:<p></p>[tex]\begin{gathered} \log_3\left((1)/(4)\right)=\log_3(1)-\log_3(4)=0-\log_3(4) \\ \\ \Rightarrow-\log_3(4) \end{gathered}$

Now, we use the change of basis formula:

$\log_c(a)=(\log_d(a))/(\log_d(c))$

When changing for a basis d greater than zero and not equal to 1.

With this, we have that

$-\log_3(4)=-(\log_(10)(4))/(\log_(10)(3))$

Applying the property:

$\log_c(a^b)=b\cdot\log_c(a)$

and knowing that 4 = 2², we get

$-(2\log_(10)(2))/(\log_(10)(3))$

We chose base 10 log because we know the following values for:

$\begin{gathered} \log_(10)(2)\approx0.3010 \\ \\ \log_(10)(3)\approx0.477 \end{gathered}$

Hence the approximation for what we want is

$\log_3\left((1)/(4)\right)\approx-(2\cdot0.3010)/(0.477)=-1.262$

This is the answer we're looking for.

0 Comments

Your comment on this question:

Your answer

1 Answer

0 Comments

Your comment on this answer:

Other Questions