Answer :
To find the inverse of the function [tex]\( f(x) = 7 \log_6(x - 4) + 5 \)[/tex], we follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 7 \log_6(x - 4) + 5 \][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = 7 \log_6(y - 4) + 5 \][/tex]
3. Isolate the logarithmic term:
- First, subtract 5 from both sides:
[tex]\[ x - 5 = 7 \log_6(y - 4) \][/tex]
- Next, divide both sides by 7:
[tex]\[ \frac{x - 5}{7} = \log_6(y - 4) \][/tex]
4. Rewrite the logarithmic equation in its exponential form to solve for [tex]\( y \)[/tex]:
[tex]\[ 6^{\frac{x - 5}{7}} = y - 4 \][/tex]
5. Isolate [tex]\( y \)[/tex] by adding 4 to both sides:
[tex]\[ y = 6^{\frac{x - 5}{7}} + 4 \][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = 6^{\frac{x - 5}{7}} + 4 \][/tex]
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 7 \log_6(x - 4) + 5 \][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = 7 \log_6(y - 4) + 5 \][/tex]
3. Isolate the logarithmic term:
- First, subtract 5 from both sides:
[tex]\[ x - 5 = 7 \log_6(y - 4) \][/tex]
- Next, divide both sides by 7:
[tex]\[ \frac{x - 5}{7} = \log_6(y - 4) \][/tex]
4. Rewrite the logarithmic equation in its exponential form to solve for [tex]\( y \)[/tex]:
[tex]\[ 6^{\frac{x - 5}{7}} = y - 4 \][/tex]
5. Isolate [tex]\( y \)[/tex] by adding 4 to both sides:
[tex]\[ y = 6^{\frac{x - 5}{7}} + 4 \][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = 6^{\frac{x - 5}{7}} + 4 \][/tex]
