Answer :
To solve the equation \( 8^x = 26 \) for \( x \), we will use logarithms and the change-of-base theorem. Let's go through the solution step by step.
1. Logarithmic Form:
The solution \( x \) is the power to which 8 must be raised to get 26. In logarithmic form, we express this as:
[tex]\[ x = \log_8 26 \][/tex]
So the first form of the solution is:
[tex]\[ \log_8 26 \][/tex]
2. Using the Common Logarithm (base 10) and the Change-of-Base Theorem:
The change-of-base theorem states that for any positive numbers \( a \), \( b \), and \( c \) (with \( a \neq 1 \) and \( b \neq 1 \)):
[tex]\[ \log_b a = \frac{\log_c a}{\log_c b} \][/tex]
Using this theorem with common logarithms (base 10), we get:
[tex]\[ x = \frac{\log 26}{\log 8} \][/tex]
So the second form of the solution is:
[tex]\[ \frac{\log 26}{\log 8} \][/tex]
3. Using the Natural Logarithm (base \( e \)):
Similarly, we can use the natural logarithm (base \( e \)) as well:
[tex]\[ x = \frac{\ln 26}{\ln 8} \][/tex]
So the third form of the solution is:
[tex]\[ \frac{\ln 26}{\ln 8} \][/tex]
Therefore, the exact solutions for \( x \) in the equation \( 8^x = 26 \) can be written in three forms:
1. \(\log_8 26\)
2. \(\frac{\log 26}{\log 8}\)
3. \(\frac{\ln 26}{\ln 8}\)
These represent the three equivalent ways to express the solution.
1. Logarithmic Form:
The solution \( x \) is the power to which 8 must be raised to get 26. In logarithmic form, we express this as:
[tex]\[ x = \log_8 26 \][/tex]
So the first form of the solution is:
[tex]\[ \log_8 26 \][/tex]
2. Using the Common Logarithm (base 10) and the Change-of-Base Theorem:
The change-of-base theorem states that for any positive numbers \( a \), \( b \), and \( c \) (with \( a \neq 1 \) and \( b \neq 1 \)):
[tex]\[ \log_b a = \frac{\log_c a}{\log_c b} \][/tex]
Using this theorem with common logarithms (base 10), we get:
[tex]\[ x = \frac{\log 26}{\log 8} \][/tex]
So the second form of the solution is:
[tex]\[ \frac{\log 26}{\log 8} \][/tex]
3. Using the Natural Logarithm (base \( e \)):
Similarly, we can use the natural logarithm (base \( e \)) as well:
[tex]\[ x = \frac{\ln 26}{\ln 8} \][/tex]
So the third form of the solution is:
[tex]\[ \frac{\ln 26}{\ln 8} \][/tex]
Therefore, the exact solutions for \( x \) in the equation \( 8^x = 26 \) can be written in three forms:
1. \(\log_8 26\)
2. \(\frac{\log 26}{\log 8}\)
3. \(\frac{\ln 26}{\ln 8}\)
These represent the three equivalent ways to express the solution.
