Question

Solve the logarithmic equation.
log 4 x plus log 5 x equals log 720
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Part 1
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​(Simplify your answer. Type an exact​ answer, using e as​ needed.)

Answer 1

To solve the logarithmic equation \( \log_4 x + \log_5 x = \log_{10} 720 \), we can first rewrite the logarithms in terms of natural logarithms using the change of base formula:

\[
\log_a b = \frac{\ln b}{\ln a}
\]

Applying this to each term in the equation:

\[
\log_4 x = \frac{\ln x}{\ln 4}
\]
\[
\log_5 x = \frac{\ln x}{\ln 5}
\]

Substituting these into the equation gives us:

\[
\frac{\ln x}{\ln 4} + \frac{\ln x}{\ln 5} = \log_{10} 720
\]

Next, we can factor \( \ln x \) out of the left side:

\[
\ln x \left( \frac{1}{\ln 4} + \frac{1}{\ln 5} \right) = \log_{10} 720
\]

We can simplify the expression inside the parentheses. Let's find a common denominator, which would be \( \ln 4 \cdot \ln 5 \):

\[
\frac{\ln 5 + \ln 4}{\ln 4 \cdot \ln 5} = \frac{\ln(4 \cdot 5)}{\ln 4 \cdot \ln 5} = \frac{\ln 20}{\ln 4 \cdot \ln 5}
\]

Thus, we have:

\[
\ln x \cdot \frac{\ln 20}{\ln 4 \cdot \ln 5} = \log_{10} 720
\]

To find \( \ln x \), we need to isolate it:

\[
\ln x = \frac{\log_{10} 720 \cdot \ln 4 \cdot \ln 5}{\ln 20}
\]

Now we can compute \( \log_{10} 720 \). Using the identity \( \log_{10} a + \log_{10} b = \log_{10} (a \cdot b) \):

\[
\log_{10} 720 = \log_{10}(72 \cdot 10) = \log_{10} 72 + 1
\]

Calculating \( \log_{10} 72 \):

- \( 72 = 2^3 \cdot 3^2 \), so:

\[
\log_{10} 72 = 3 \log_{10} 2 + 2 \log_{10} 3
\]

We know approximate values for \( \log_{10} 2 \approx 0.301 \) and \( \log_{10} 3 \approx 0.477 \):

\[
\log_{10} 72 \approx 3(0.301) + 2(0.477) = 0.903 + 0.954 = 1.857
\]

Thus:

\[
\log_{10} 720 \approx 1.857 + 1 = 2.857
\]

Substituting this back, we can find \( \ln x \):

Let's summarize:
- We now need the exact values of \( \ln 4 \) and \( \ln 5 \):
- \( \ln 4 = 2 \ln 2 \) (approximately \( 2 \cdot 0.693 = 1.386 \))
- \( \ln 5 \) can be found similarly if needed.

We can finalize \( x \) using:

\[
x = e^{\frac{\log_{10} 720 \cdot \ln 4 \cdot \ln 5}{\ln 20}}
\]

This expression will yield the exact answer for \( x \). Let's simplify \( \frac{\log_{10} 720}{\ln 20} = \frac{2.857}{\ln 20} \).

Using calculators for the logs can provide the numerical value directly if you need it.

It's essential to ensure that the calculation follows through. Confirm all values of constants or approximations you use from logarithm tables or calculators as required.