Answer :
Let's solve the exponential equation \(6^x = 30\) and express the exact solution in three different logarithmic forms, similar to how \(\log_5 9\) can be expressed as \(\frac{\log 9}{\log 5}\) or \(\frac{\ln 9}{\ln 5}\).
1. Exact log form:
The logarithmic form of the solution involves using the base 6 logarithm, representing the power to which 6 must be raised to get 30:
[tex]\[ x = \log_6 30 \][/tex]
2. Change-of-base formula (using common logarithm \(\log\)):
We can apply the change-of-base theorem, which allows us to convert the logarithm into any base. The common logarithm (base 10) is often used:
[tex]\[ x = \frac{\log 30}{\log 6} \][/tex]
3. Change-of-base formula (using natural logarithm \(\ln\)):
We can also use the natural logarithm (base \(e\)) to express the solution:
[tex]\[ x = \frac{\ln 30}{\ln 6} \][/tex]
Therefore, the exact solutions for the equation \(6^x = 30\) in the three different forms are:
1. \(\log_6 30\)
2. \(\frac{\log 30}{\log 6}\)
3. \(\frac{\ln 30}{\ln 6}\)
Hence, filling the exact solution forms into the provided form looks like:
For the given equation, the exact solutions in three forms are [tex]\(\log_6 30\)[/tex], [tex]\(\frac{\log 30}{\log 6}\)[/tex], and [tex]\(\frac{\ln 30}{\ln 6}\)[/tex].
1. Exact log form:
The logarithmic form of the solution involves using the base 6 logarithm, representing the power to which 6 must be raised to get 30:
[tex]\[ x = \log_6 30 \][/tex]
2. Change-of-base formula (using common logarithm \(\log\)):
We can apply the change-of-base theorem, which allows us to convert the logarithm into any base. The common logarithm (base 10) is often used:
[tex]\[ x = \frac{\log 30}{\log 6} \][/tex]
3. Change-of-base formula (using natural logarithm \(\ln\)):
We can also use the natural logarithm (base \(e\)) to express the solution:
[tex]\[ x = \frac{\ln 30}{\ln 6} \][/tex]
Therefore, the exact solutions for the equation \(6^x = 30\) in the three different forms are:
1. \(\log_6 30\)
2. \(\frac{\log 30}{\log 6}\)
3. \(\frac{\ln 30}{\ln 6}\)
Hence, filling the exact solution forms into the provided form looks like:
For the given equation, the exact solutions in three forms are [tex]\(\log_6 30\)[/tex], [tex]\(\frac{\log 30}{\log 6}\)[/tex], and [tex]\(\frac{\ln 30}{\ln 6}\)[/tex].
