# 13.3: Exercises

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## Exercise $$\PageIndex{1}$$

Graph the following functions with the calculator.

1. $$y=5^x$$
2. $$y=1.01^x$$
3. $$y=\left (\dfrac 1 3 \right )^x$$
4. $$y=0.97^x$$
5. $$y=3^{-x}$$
6. $$y=\left (\dfrac 1 3 \right )^{-x}$$
7. $$y=e^{x^2}$$
8. $$y=0.01^x$$
9. $$y=1^x$$
10. $$y=e^{x}+1$$
11. $$y=\dfrac{e^x-e^{-x}}{2}$$
12. $$y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$$

The last two functions are known as the hyperbolic sine, $$\sinh(x)=\dfrac{e^x-e^{-x}}{2}$$, and the hyperbolic tangent, $$\tanh(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$$. Recall that the hyperbolic cosine $$\cosh(x)=\dfrac{e^x+e^{-x}}{2}$$ was already graphed in example hyperbolic-cosine.

1. same as c) since $$y=\left(\dfrac{1}{3}\right)^{x}=3^{-x}$$

## Exercise $$\PageIndex{2}$$

Graph the given function. Describe how the graph is obtained by a transformation from the graph of an exponential function $$y=b^x$$ (for appropriate base $$b$$).

1. $$y=0.1\cdot 4^x$$
2. $$y=3\cdot 2^x$$
3. $$y=(-1)\cdot 2^x$$
4. $$y=0.006\cdot 2^x$$
5. $$y=e^{-x}$$
6. $$y=e^{-x}+1$$
7. $$y=(\dfrac 1 2)^{x}+3$$
8. $$y=2^{x-4}$$
9. $$y=2^{x+1}-6$$
1. $$y=4^{x}$$ is compressed towards the x-axis by the factor $$0.1$$
2. $$y = 2^x$$ stretched away from $$x$$-axis
3. $$y = 2^x$$ reflected about the $$x$$-axis
4. $$y = 2^x$$ compressed towards the $$x$$-axis
5. $$y = e^x$$ reflected about the $$y$$-axis
6. $$y = e^x$$ reflected about the $$y$$-axis and shifted up by $$1$$
7. $$y=\left(\dfrac{1}{2}\right)^{x}$$ shifted up by $$3$$
8. $$y = 2^x$$ shifted to the right by $$4$$
9. $$y = 2^x$$ shifted to the left by $$1$$ and down by $$6$$

## Exercise $$\PageIndex{3}$$

Use the definition of the logarithm to write the given equation as an equivalent logarithmic equation.

1. $$4^2=16$$
2. $$2^{8}=256$$
3. $$e^x=7$$
4. $$10^{-1}=0.1$$
5. $$3^x=12$$
6. $$5^{7\cdot x}=12$$
7. $$3^{2a+1}=44$$
8. $$\left(\dfrac{1}{2}\right)^{\frac{x}{h}}=30$$
1. $$\log _{4}(16)=2$$
2. $$\log _{2}(256)=8$$
3. $$\ln (7)=x$$
4. $$\log (0.1)=-1$$
5. $$\log _{3}(12)=x$$
6. $$\log _{5}(12)=7 x$$
7. $$\log _{3}(44)=2 a+1$$
8. $$\log _{\frac{1}{2}}(30)=\dfrac{x}{h}$$

## Exercise $$\PageIndex{4}$$

Evaluate the following expressions without using a calculator.

1. $$\log_7(49)$$
2. $$\log_3(81)$$
3. $$\log_{2}(64)$$
4. $$\log_{50}(2,500)$$
5. $$\log_2(0.25)$$
6. $$\log(1,000)$$
7. $$\ln(e^4)$$
8. $$\log_{13}(13)$$
9. $$\log(0.1)$$
10. $$\log_6 \left (\dfrac 1 {36} \right)$$
11. $$\ln(1)$$
12. $$\log_{\frac 1 2}(8)$$
1. $$2$$
2. $$4$$
3. $$6$$
4. $$2$$
5. $$−2$$
6. $$3$$
7. $$4$$
8. $$1$$
9. $$−1$$
10. $$−2$$
11. $$0$$
12. $$−3$$

## Exercise $$\PageIndex{5}$$

Using a calculator, approximate the following expressions to the nearest thousandth.

1. $$\log_3(50)$$
2. $$\log_3(12)$$
3. $$\log_{17}(0.44)$$
4. $$\log_{0.34}(200)$$
1. $$3.561$$
2. $$2.262$$
3. $$−0.290$$
4. $$−4.911$$

## Exercise $$\PageIndex{6}$$

State the domain of the function $$f$$ and sketch its graph.

1. $$f(x)=\log(x)$$
2. $$f(x)=\log(x+7)$$
3. $$f(x)=\ln(x+5)-1$$
4. $$f(x)=\ln(3x-6)$$
5. $$f(x)=2\cdot \log(x+4)$$
6. $$f(x)=-4\cdot\log(x+2)$$
7. $$f(x)=\log_{3}(x-5)$$
8. $$f(x)=-\ln(x-1)+3$$
9. $$f(x)=\log_{0.4}(x)$$
10. $$f(x)=\log_{3}(-5x)-2$$
11. $$f(x)=\log|x|$$
12. $$f(x)=\log|x+2|$$