### Learning Outcomes

- Combine the product, power, and quotient rules to condense logarithmic expressions
- Simplify expressions using the Product Property of Exponents
- Simplifying expressions using the Quotient Property of Exponents
- Simplify expressions using the Power Property of Exponents
- Use logarithms to solve exponential equations whose terms cannot be rewritten with the same base
- Solve exponential equations of the form [latex]y=Ae^{kt}[/latex] for [latex]t[/latex]

In the Integrals, Exponential Functions, and Logarithms and Exponential Growth and Decay sections, we will look more into differentiating and integrating exponential and logarithmic functions. Here we will review condensing logarithmic expressions, using rules of exponents, and how to solve exponential equations.

## Condense Logarithmic Expressions

### A General Note: The Product Rule for Logarithms

The **product rule for logarithms** can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.

[latex]{\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)\text{ for }b>0[/latex]

### A General Note: The Quotient Rule for Logarithms

The **quotient rule for logarithms** can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.

[latex]{\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}M-{\mathrm{log}}_{b}N[/latex]

### A General Note: The Power Rule for Logarithms

The **power rule for logarithms** can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

[latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M[/latex]

We can use the rules of logarithms to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined.

### How To: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm

- Apply the power property first. Identify terms that are products of factors and a logarithm and rewrite each as the logarithm of a power.
- From left to right, apply the product and quotient properties. Rewrite sums of logarithms as the logarithm of a product and differences of logarithms as the logarithm of a quotient.

### Example: Condensing Logarithmic Expressions

Use the power rule for logs to rewrite [latex]4\mathrm{ln}\left(x\right)[/latex] as a single logarithm with a leading coefficient of 1.

### Try It

Use the power rule for logs to rewrite [latex]2{\mathrm{log}}_{3}4[/latex] as a single logarithm with a leading coefficient of 1.

In our next few examples, we will use a combination of logarithm rules to condense logarithms.

### Example: Condensing Logarithmic Expressions

Write [latex]{\mathrm{log}}_{3}\left(5\right)+{\mathrm{log}}_{3}\left(8\right)-{\mathrm{log}}_{3}\left(2\right)[/latex] as a single logarithm.

### Try It

## Use Rules of Exponents

### A General Note: The Product Rule of Exponents

For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that

### Example: Using the Product Rule

Write each of the following products with a single base. Do not simplify further.

- [latex]{t}^{5}\cdot {t}^{3}[/latex]
- [latex]\left(-3\right)^{5}\cdot \left(-3\right)[/latex]
- [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex]

### A General Note: The Quotient Rule of Exponents

For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], such that [latex]m>n[/latex], the quotient rule of exponents states that

### Example: Using the Quotient Rule

Write each of the following products with a single base. Do not simplify further.

- [latex]\dfrac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex]
- [latex]\dfrac{{t}^{23}}{{t}^{15}}[/latex]
- [latex]\dfrac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex]

### A General Note: The Power Rule of Exponents

For any real number [latex]a[/latex] and positive integers [latex]m[/latex] and [latex]n[/latex], the power rule of exponents states that

Write each of the following products with a single base. Do not simplify further.

- [latex]{\left({x}^{2}\right)}^{7}[/latex]
- [latex]{\left({\left(2t\right)}^{5}\right)}^{3}[/latex]
- [latex]{\left({\left(-3\right)}^{5}\right)}^{11}[/latex]

### Try It

https://ohm.lumenlearning.com/multiembedq.php?id=93370&theme=oea&iframe_resize_id=mom80

https://ohm.lumenlearning.com/multiembedq.php?id=93399&theme=oea&iframe_resize_id=mom90

## Solve Exponential Equations

One technique that is used to solve exponential equations is to take the logarithm of each side of the equation.

### How To: Given an exponential equation, Solve By Taking the Logarithm of Each Side

- Apply the logarithm to both sides of the equation.
- If one of the terms in the equation has base 10, use the common logarithm.
- If none of the terms in the equation has base 10, use the natural logarithm.

- Use the rules of logarithms to solve for the unknown.

### Example: Solving Exponential Equations

Solve [latex]{5}^{x+2}={4}^{x}[/latex].

### Try It

Solve [latex]{2}^{x}={3}^{x+1}[/latex].

### Try It

One common type of exponential equations are those with base *e*. This constant occurs again and again in nature, mathematics, science, engineering, and finance. When we have an equation with a base *e* on either side, we can use the **natural logarithm** to solve it.

### How To: Given an equation of the form [latex]y=A{e}^{kt}[/latex], solve for [latex]t[/latex]

- Divide both sides of the equation by
*A*. - Apply the natural logarithm to both sides of the equation.
- Divide both sides of the equation by
*k*.

### Example: Solving Exponential Equations with Base [latex]e[/latex]

Solve [latex]100=20{e}^{2t}[/latex].

### Try It

Solve [latex]3{e}^{0.5t}=11[/latex].