Evaluate which of the following expressions is equivalent to

Question

log
x
log
4
.(1 point)
Responses

log
x
x
log
4
Start Fraction logarithm subscript x baseline x over logarithm 4 End Fraction

log
3
x
log
3
4
Start Fraction logarithm subscript 3 baseline x over logarithm subscript 3 baseline 4 End Fraction

log
x
4
logarithm subscript x baseline 4

ln
x
ln
4

Answer 1

To evaluate which expression is equivalent to \(\frac{\log x}{\log 4}\), we will use the change of base formula for logarithms, which states:

\[ \log_b a = \frac{\log_k a}{\log_k b} \]

for any base \(k\).

Using this, we can rewrite \(\frac{\log x}{\log 4}\) as:

\[ \frac{\log x}{\log 4} = \log_4 x \]

Now, let's analyze each option provided:

1. \( \frac{\log_x x}{\log 4} \)

   • Here, \(\log_x x = 1\) (since any log of a number to its own base is 1).
   • So, \( \frac{1}{\log 4} \) is not equivalent to \(\frac{\log x}{\log 4}\).

2. \( \frac{\log_3 x}{\log_3 4} \)

   • This can be rewritten using the change of base formula: \(\frac{\log_3 x}{\log_3 4} = \log_4 x\), which is equivalent to \(\frac{\log x}{\log 4}\). This is a potential match.

3. \( \log_x 4 \)

   • This is not equivalent to \(\frac{\log x}{\log 4}\).

4. \( \frac{\ln x}{\ln 4} \)

   • This can also be rewritten using the change of base formula: \(\frac{\ln x}{\ln 4} = \log_4 x\), which is the same as \(\frac{\log x}{\log 4}\) since \(\log\) and \(\ln\) are simply logs with different bases. This is also a potential match.

Summarizing:

• The expressions that are equivalent to \(\frac{\log x}{\log 4}\) are:
  • \( \frac{\log_3 x}{\log_3 4} \)
  • \( \frac{\ln x}{\ln 4} \)

Therefore, both options 2 and 4 are correct equivalences of \(\frac{\log x}{\log 4}\).