Properties of Logarithms Lesson

In the last lesson, we introduced the concept of a logarithm. In this lesson, we will go deeper into the topic of logarithms. Let's begin by learning about the product rule for logarithms.

Product Rule for Logarithms

If x, y, and b are positive real numbers, where b ≠ 1, then:
log_b(xy) = log_b(x) + log_b(y)
Let's take a look at a few examples.
Note: All variables in our examples represent positive real numbers.
Example 1: Expand each logarithm.
log₁₀(3y)
Using the product rule for logarithms, we can expand our logarithm:
log₁₀(3y) = log₁₀(3) + log₁₀(y)
Example 2: Condense each expression into a single logarithm.
log₇(5) + log₇(3)
We can use the product rule for logarithms to condense our expression into a single logarithm.
log₇(5) + log₇(3) = log₇(5 • 3) = log₇(15)

Quotient Rule for Logarithms

In addition to the product rule for logarithms, we also have a quotient rule for logarithms. If x, y, and b are positive real numbers, where b ≠ 1, then: $$\text{log}_{b}\left(\frac{x}{y}\right)=\text{log}_{b}(x) - \text{log}_{b}(y)$$ Let's look at a few examples.
Example 3: Expand each logarithm. $$\text{log}_{12}\left(\frac{11}{5}\right)$$ Using the quotient rule for logarithms, we can expand our logarithm: $$\text{log}_{12}\left(\frac{11}{5}\right)=\text{log}_{12}(11) - \text{log}_{12}(5)$$ Example 4: Condense each expression into a single logarithm. $$\text{log}_{8}(x) - \text{log}_{8}(z)$$ We can use the quotient rule for logarithms to condense our expression into a single logarithm. $$\text{log}_{8}(x) - \text{log}_{8}(z)=\text{log}_{8}\left(\frac{x}{z}\right)$$

Power Rule for Logarithms

We also have a rule for logarithms that deals with powers. If x and b are positive real numbers, where b ≠ 1, and if r is any real number, then:
log_b(x^r) = rlog_b(x)
This rule allows us to take the exponent out of the argument. In other words, the logarithm of a number that is raised to a power is equal to the exponent multiplied by the logarithm of the number. Let's look at an example.
Example 5: Use the power rule to rewrite each logarithm.
log₃(x⁹)
Using the power rule for logarithms, we can rewrite our logarithm:
log₃(x⁹) = 9log₃(x)

Special Properties of Logarithms

Lastly, we want to learn two special properties of logarithms.
If b > 0 and b ≠ 1, then:
b^log_b(x) = x, x > 0
log_b(b^x) = x
Let's look at a few examples.
Example 6: Evaluate each.
log₁₁(11⁴)
Using our above rules, we can evaluate our problem as:
log₁₁(11⁴) = 4
Example 7: Evaluate each.
10^{log₁₀(19)}
Using our above rules, we can evaluate our problem as:
10^{log₁₀(19)} = 19

Using the Properties of Logarithms to write alternative forms of logarithmic expressions

Now, we will put everything together and use our properties of logarithms to rewrite logarithmic expressions. Being able to apply the properties of logarithms will be important for solving equations that require logarithms. Let's look at a few examples.
Example 8: Expand each logarithm.
log₈(x³ • y)⁵
Let's start by moving the exponent of 5 out in front of the logarithm:
5log₈(x³ • y)
Now, let's use our product rule for logarithms to expand:
5(log₈(x³) + log₈(y))
5log₈(x³) + 5log₈(y)
Lastly, we can move the exponent of 3 out in front:
(3 • 5)log₈(x) + 5log₈(y)
15log₈(x) + 5log₈(y)
Example 9: Condense each expression into a single logarithm. $$\text{log}_{5}(z) + \frac{\text{log}_{5}(x)}{2}+ \frac{\text{log}_{5}(y)}{2}$$ Let's rewrite division by 2 as multiplication by 1/2: $$\text{log}_{5}(z) + \frac{1}{2}\text{log}_{5}(x) + \frac{1}{2}\text{log}_{5}(y)$$ Using the power rule for logarithms, we can rewrite our problem as: $$\text{log}_{5}(z) + \text{log}_{5}(x^{\frac{1}{2}}) + \text{log}_{5}(y^{\frac{1}{2}})$$ When we raise a number to the power of 1/2, this is the same as taking the square root of the number: $$\text{log}_{5}(z) + \text{log}_{5}(\sqrt{x}) + \text{log}_{5}(\sqrt{y})$$ Now, we can wrap up the problem by using our product rule for logarithms: $$\text{log}_{5}(z\sqrt{xy})$$

Deriving the Product Rule for Logarithms

$$\text{log}_b(x)=m$$ $$\text{log}_b(y)=n$$ Write each in exponential form: $$b^m=x$$ $$b^n=y$$ Think about the product xy: $$xy=b^m \cdot b^n$$ $$xy=b^{m + n}$$ Convert into logarithmic form: $$\text{log}_b(xy)=m + n$$ Substitute, for m and n: $$\text{log}_b(xy)=\text{log}_b(x) + \text{log}_b(y)$$

Deriving the Quotient Rule for Logarithms

$$\text{log}_b(x)=m$$ $$\text{log}_b(y)=n$$ Write each in exponential form: $$b^m=x$$ $$b^n=y$$ Think about the quotient x/y: $$\frac{x}{y}=\frac{b^m}{b^n}$$ $$\frac{x}{y}=b^{m - n}$$ Convert into logarithmic form: $$\text{log}_b\left(\frac{x}{y}\right)=m - n$$ Substitute, for m and n: $$\text{log}_b\left(\frac{x}{y}\right)=\text{log}_b(x) - \text{log}_b(y)$$

Deriving the Power Rule for Logarithms

$$\text{log}_b(x)=m$$ Write in exponential form: $$b^m=x$$ Raise each side to the power r: $$\left(b^m\right)^r=x^r$$ Power to Power Rule: $$\left(b^{mr}\right)=x^r$$ Convert into logarithmic form: $$\text{log}_b(x^r)=mr$$ Substitute for m: $$\text{log}_b(x^r)=\text{log}_b(x) \cdot r$$ Rearrange: $$\text{log}_b(x^r)=r \cdot \text{log}_b(x)$$

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