To solve the equation [tex]\(\frac{\log x}{\log 4} = \frac{\log 64}{\log 256}\)[/tex], we proceed with the following steps:
1. Understand the Right-Hand Side of the Equation:
Calculate [tex]\(\frac{\log 64}{\log 256}\)[/tex]:
- The logarithm (log) values of 64 and 256 are:
[tex]\[
\log 64 = 4.1588830833596715
\][/tex]
[tex]\[
\log 256 = 5.545177444479562
\][/tex]
- Hence,
[tex]\[
\frac{\log 64}{\log 256} = \frac{4.1588830833596715}{5.545177444479562} = 0.75
\][/tex]
2. Equating and Solving for [tex]\(\log x\)[/tex]:
Since [tex]\(\frac{\log x}{\log 4} = 0.75\)[/tex], we can isolate [tex]\(\log x\)[/tex] as follows:
- First, know that [tex]\(\log 4 = 1.3862943611198906\)[/tex].
- Substitute:
[tex]\[
\frac{\log x}{1.3862943611198906} = 0.75
\][/tex]
- Solving for [tex]\(\log x\)[/tex]:
[tex]\[
\log x = 0.75 \times 1.3862943611198906
\][/tex]
[tex]\[
\log x = 1.0397207708399179
\][/tex]
3. Exponentiating to Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], exponentiate the result:
- If [tex]\(\log x = 1.0397207708399179\)[/tex], then:
[tex]\[
x = e^{1.0397207708399179}
\][/tex]
[tex]\[
x = 2.82842712474619
\][/tex]
Thus, the value of [tex]\(x\)[/tex] that satisfies the given equation is [tex]\(2.82842712474619\)[/tex].
