Question

1. rewrite the equation $6^2 = 36$ in logarithmic form

$\square=\square$

1. find the inverse of $f(x) = 0.3^x$.

$f^{-1}(x)=\square$

1. find the inverse of $y = \log_2 x$.

$y=\square$

Explanation:

Question 13

Step1: Recall the definition of logarithms. For \( a^b = c \), the logarithmic form is \( \log_a c = b \).

Here, \( a = 6 \), \( b = 2 \), and \( c = 36 \).

Step2: Apply the definition to the given equation \( 6^2 = 36 \).

So the logarithmic form is \( \log_6 36 = 2 \).

Step1: Let \( y = f(x) = 0.3^x \). To find the inverse, we first swap \( x \) and \( y \).

So we get \( x = 0.3^y \).

Step2: Convert the exponential equation to logarithmic form. Recall that if \( a^b = c \), then \( \log_a c = b \). Here, \( a = 0.3 \), \( b = y \), and \( c = x \). So \( y = \log_{0.3} x \).

Step3: Replace \( y \) with \( f^{-1}(x) \).

So \( f^{-1}(x) = \log_{0.3} x \) (or we can also write it using natural logarithm or common logarithm, but the form with base \( 0.3 \) is direct).

Step1: Let \( y = \log_2 x \). To find the inverse, we first convert the logarithmic equation to exponential form. Recall that \( \log_a b = c \) is equivalent to \( a^c = b \).

So from \( y = \log_2 x \), we get \( 2^y = x \).

Step2: Swap \( x \) and \( y \) to find the inverse function.

After swapping, we have \( 2^x = y \).

Step3: Replace \( y \) with the inverse function notation.

So the inverse of \( y = \log_2 x \) is \( y = 2^x \).

Answer:

\( \log_{6} 36 = 2 \)

Question 14