Question

question 23
$y = 5^x - 9$
a) $y = \log_6 -4x$
b) $y = \log_{\frac{1}{2}} (x - 10)$
c) $y = \log_2 x^5$
d) $y = \log_5 (x + 9)$
options: c, b, d, a

Explanation:

Step1: Recall inverse of exponential function

To find the inverse of \( y = 5^x - 9 \), first, we swap \( x \) and \( y \), so we get \( x = 5^y - 9 \).

Step2: Solve for \( y \)

Add 9 to both sides: \( x + 9 = 5^y \).
Now, convert the exponential equation to a logarithmic equation. Remember that if \( a^b = c \), then \( \log_a c = b \). Here, \( a = 5 \), \( b = y \), and \( c = x + 9 \). So, \( y=\log_{5}(x + 9) \).

Answer:

D. \( y=\log_{5}(x + 9) \)