Question Type 2: Drawing logarithmic functions as inverses of the exponential functions Exercises

Sketch $y=5^{2x}$ and its inverse $y=\frac{1}{2}\log_5(x)$. Label asymptotes, the line $y=x$, the points $(0,1)$ and $(\frac{1}{2},5)$, and their images.

Sketch $y=3^{-x}$ and its inverse $y=-\log_3(x)$. Label asymptotes, line $y=x$, and images of $(0,1)$ and $(1,\frac{1}{3})$.

Sketch $y=e^{x-1}$ and its inverse $y=\ln(x)+1$. Label the asymptotes, the line $y=x$, and the images of $(1,1)$ and $(2,e)$.

Sketch $y=5^{x+2}-1$ and its inverse. State the equation of the inverse before sketching.

Sketch the graphs of $y = e^x$ and its inverse $y = \ln(x)$ on the same axes. Label the asymptotes, the line $y=x$, and the points corresponding to $(0,1)$ and $(1,e)$ with their reflections.

Sketch the graph of $y=\log_{1/2}(x)$ and its inverse $y=(1/2)^x$.

Label the asymptotes, the line $y=x$, and the points $(1,0)$ and $(2,-1)$ along with their reflections.

Consider the function $f(x) = \frac{1}{3}e^{x-1}$.

Find the equation of the inverse function, $f^{-1}(x)$. Hence, sketch the graphs of $f$ and $f^{-1}$ on the same set of axes, indicating the relationship between them.

On one set of axes, sketch $y=2\ln(x)$ and its inverse $y=e^{x/2}$. Label the asymptotes, the line $y=x$, and the points $(1,0)$ and $(e,2)$ and their images.

Sketch $y=5^x$ and its inverse $y=\log_5(x)$, labelling the asymptotes, the line $y=x$, and the key points $(0,1)$ and $(1,5)$ with their reflections.

Sketch $y=\log_3(x-4)+2$ and its inverse. Determine the inverse explicitly before sketching.

On one set of axes, sketch $y=2^x+3$ and its inverse $y=\log_2(x-3)$. Label the asymptotes, the line $y=x$, and key points.

Find the expression for the inverse function of $y=\frac{\ln(x)}{2\ln(5)}$. Sketch both functions on the same set of axes. Label asymptotes, the line $y=x$, and the points $(1,0)$ and $(5,\frac{1}{2})$ along with their reflections.