# 5.3 Pearson Online HW Solutions

## Question 1. Graph the equation. $y=\log_3 x$

<details> <summary> Solution: </summary>

| $x = 3^y$ | $y$ |
|-------------------|------|
| 1 | 0 |
| 3 | 1 |
| 9 | 2 |
| 27 | 3 |
| 81 | 4 |
| $\frac{1}{3}$ | -1 |
| $\frac{1}{9}$ | -2 |

| $x$ | $y = \log_3 x$ |
|------------|----------------|
| 1 | 0 |
| 3 | 1 |
| 9 | 2 |
| 27 | 3 |
| 81 | 4 |
| $\frac{1}{3}$ | -1 |
| $\frac{1}{9}$ | -2 |

| Property | Value |
|--------------------|------------------------------|
| Domain | $x > 0$ |
| Range | $(-\infty, \infty)$ |
| $x$-intercept | $(1, 0)$ |
| Asymptote | Vertical asymptote at $x = 0$|

The graph of $y = \log_3 x$ is a logarithmic curve that passes through $(1,0)$, increases slowly for $x > 1$, and approaches negative infinity as $x \to 0^+$.


</details>

## Question 2. Find the logarithm. $\log_8 4096$

<details> <summary> Solution: </summary>

This means we can convert it to the exponential equation:

$$
8^x = 4096
$$

Now, find the value of $x$ that makes this true.

Since $8^4 = 4096$, we have:

$$
x = 4
$$

Thus:

$$
\log_8 4096 = 4
$$

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## Question 3. Find the logarithm. $\log_{10} 0.01$

<details> <summary> Solution: </summary>

Let $\log_{10} 0.01 = x$.

This means we can convert it to the exponential equation:

$$
10^x = 0.01
$$

Now, find the value of $x$ that makes this true.

Since $10^{-2} = 0.01$, we have:

$$
x = -2
$$

Thus:

$$
\log_{10} 0.01 = -2
$$

</details>

## Question 4. Find the logarithm. $\log_5 \left(\dfrac{1}{125}\right)$

<details> <summary> Solution: </summary>

Let $\log_5 \left(\dfrac{1}{125}\right) = x$.

This means we can convert it to the exponential equation:

$$
5^x = \dfrac{1}{125}
$$

Now, find the value of $x$ that makes this true.

Since $5^{-3} = \dfrac{1}{125}$, we have:

$$
x = -3
$$

Thus:

$$
\log_5 \dfrac{1}{125} = -3
$$

</details>

## Question 5. Find the logarithm. $\log(10)$

<details> <summary> Solution: </summary>

Let $\log_{10} 10 = x$.

This means we can convert it to the exponential equation:

$$
10^x = 10
$$

Now, find the value of $x$ that makes this true.

Since $10^1 = 10$, we have:

$$
x = 1
$$

Thus:

$$
\log_{10} 10 = 1
$$

</details>

## Question 6. Find the logarithm. $\log_5 5^4$

<details> <summary> Solution: </summary>

Let $\log_5 5^4 = x$.

This means we can convert it to the exponential equation:

$$
5^x = 5^4
$$

Now, find the value of $x$ that makes this true.

Since the bases are the same, we can equate the exponents:

$$
x = 4
$$

Thus:

$$
\log_5 5^4 = 4
$$

</details>

## Question 7. Find the logarithm. $\log_3 \sqrt[5]{3}$

<details> <summary> Solution: </summary>

Let $\log_3 \sqrt[5]{3} = x$.

This means we can convert it to the exponential equation:

$$
3^x = \sqrt[5]{3}
$$

Now, rewrite $\sqrt[5]{3}$ as a power of 3:

$$
\sqrt[5]{3} = 3^{\frac{1}{5}}
$$

Thus, we have:

$$
3^x = 3^{\frac{1}{5}}
$$

Since the bases are the same, we can equate the exponents:

$$
x = \frac{1}{5}
$$

Therefore:

$$
\log_3 \sqrt[5]{3} = \frac{1}{5}
$$

</details>

## Question 8. Find the logarithm. $\log_{16} 2$

<details> <summary> Solution: </summary>

Let $\log_{16} 2 = x$.

This means we can convert it to the exponential equation:

$$
16^x = 2
$$

Now, rewrite $16$ as a power of $2$:

$$
16 = 2^4
$$

So we have:

$$
(2^4)^x = 2
$$

This simplifies to:

$$
2^{4x} = 2^1
$$

Since the bases are the same, we can equate the exponents:

$$
4x = 1
$$

Now, solve for $x$:

$$
x = \frac{1}{4}
$$

Therefore:

$$
\log_{16} 2 = \frac{1}{4}
$$

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## Question 9. Find the logarithm. $\log_{36} 1$

<details> <summary> Solution: </summary>

Let $\log_{36} 1 = x$.

This means we can convert it to the exponential equation:

$$
36^x = 1
$$

We know that any number raised to the power of $0$ is $1$. Thus:

$$
x = 0
$$

Therefore:

$$
\log_{36} 1 = 0
$$

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## Question 10. Find the logarithm. Do not use a calculator. $\ln \sqrt[9]{e}$

<details> <summary> Solution: </summary>

Let $\ln \sqrt[9]{e} = x$.

This means we can convert it to the exponential equation:

$$
e^x = \sqrt[9]{e}
$$

Now, rewrite $\sqrt[9]{e}$ as a power of $e$:

$$
\sqrt[9]{e} = e^{\frac{1}{9}}
$$

So we have:

$$
e^x = e^{\frac{1}{9}}
$$

Since the bases are the same, we can equate the exponents:

$$
x = \frac{1}{9}
$$

Therefore:

$$
\ln \sqrt[9]{e} = \frac{1}{9}
$$

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## Question 11. Convert to a logarithm equation. $10^2 =100$

<details> <summary> Solution: </summary>

To convert the equation $10^2 = 100$ to a logarithmic form, we write:

$$
\log_{10} 100 = 2
$$

Thus, the equivalent logarithmic equation is:

$$
\log_{10} 100 = 2
$$

</details>

## Question 12. Convert to a logarithm equation. $9^{1/2}=3$

<details> <summary> Solution: </summary>

To convert the equation $9^{1/2} = 3$ to a logarithmic form, we write:

$$
\log_9 3 = \frac{1}{2}
$$

Thus, the equivalent logarithmic equation is:

$$
\log_9 3 = \frac{1}{2}
$$

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## Question 13. Convert to a logarithm equation. $e^1=2.7183$

<details> <summary> Solution: </summary>

To convert the equation $e^1 = 2.7183$ to a logarithmic form, we write:

$$
\ln 2.7183 = 1
$$

Thus, the equivalent logarithmic equation is:

$$
\ln 2.7183 = 1
$$

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## Question 14. Convert to an exponential equation. $\log 0.01=-2$

<details> <summary> Solution: </summary>

To convert the equation $\log 0.01 = -2$ to an exponential form, we write:

$$
10^{-2} = 0.01
$$

Thus, the equivalent exponential equation is:

$$
10^{-2} = 0.01
$$

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## Question 15. Convert to an exponential equation. $\ln 75=4.3175$

<details> <summary> Solution: </summary>

To convert the equation $\ln 75 = 4.3175$ to an exponential form, we write:

$$
e^{4.3175} = 75
$$

Thus, the equivalent exponential equation is:

$$
e^{4.3175} = 75
$$

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## Question 16. Convert to an exponential equation. $x=\log_m T^4$

<details> <summary> Solution: </summary>

To convert the equation $x = \log_m T^4$ to an exponential form, we write:

$$
m^x = T^4
$$

Thus, the equivalent exponential equation is:

$$
m^x = T^4
$$

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## Question 17. Use a calculator to find the common logarithm. $\log 46$

<details> <summary> Solution: </summary>

The common logarithm of 46, denoted as $\log 46$, can be found using a calculator:

$$
\log 46 \approx 1.6628
$$

So,

$$
\log 46 \approx 1.6628
$$

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## Question 18. Use a calculator to find the common logarithm. $\log (-46)$

<details> <summary> Solution: </summary>

The common logarithm of a negative number, such as $\log(-46)$, is undefined in the real number system.

Logarithms are only defined for positive real numbers. Therefore:

$$
\log(-46) \text{ is undefined.}
$$

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## Question 19. Use a calculator to find the common logarithm. $\ln 340.9$

<details> <summary> Solution: </summary>

The natural logarithm of 340.9, denoted as $\ln 340.9$, can be found using a calculator:

$$
\ln 340.9 \approx 5.8324
$$

So,

$$
\ln 340.9 \approx 5.8324
$$

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