Lesson Objectives

- Demonstrate an understanding of logarithms
- Learn how to approximate the value of a common logarithm
- Learn how to approximate the value of a natural logarithm
- Learn how to use the change of base rule to approximate any logarithm

## How to Use the Change of Base Formula to Approximate Logarithms

### Common Logarithm - Base 10 Logarithm

In our previous dealings with the base 10 logarithm, we listed the base just like for any other. Moving forward, we no longer have to list the base for a base 10 logarithm. This is because the base 10 logarithm is used so frequently, it is known as the "common logarithm".log

_{10}(x) = log(x)

In algebra, when we see a logarithm with no base listed, it is understood that we are working with a base 10 logarithm. We can easily approximate the value of a base 10 logarithm using a calculator. Pretty much every calculator has a LOG key. This LOG key stands for base 10 logarithm. If we wanted to find log(15), we could type this into a calculator and get approximately 1.1761 (rounded to the nearest ten-thousandth). In most cases, we will need to round our answer. Check with the teacher or instructions on the assignment to determine how accurate the approximation needs to be. Let's look at an example.

Example 1: Use a calculator to approximate the answer (round to the nearest ten-thousandth).

log(41)

To approximate the value, we type in log of 41 on our calculator. We then round to the nearest ten-thousandth. Make sure to use the "≈" symbol instead of the "=" symbol. We are giving an estimate, not an exact value.

log(41) ≈ 1.6128

### Natural Logarithm - Base e Logarithm

Another commonly used logarithm is known as the "natural logarithm". The natural logarithm has a base of e. This "e" is a special number, kind of like pi "π". The number e is irrational. This means its decimal does not terminate or repeat the same pattern. The natural log derives its name from the fact that natural logs are used in many science or biology situations with growth or decay. The base e logarithm is displayed as:ln(x)

In other words:

log

_{e}(x) and ln(x) are the same.

The majority of calculators also have an LN key. This means we can also approximate the value of a natural logarithm using a calculator. Let's look at an example.

Example 2: Use a calculator to approximate the answer (round to the nearest ten-thousandth).

ln(51)

To approximate the value, we type in ln of 51 on our calculator. We then round to the nearest ten-thousandth. Make sure to use the "≈" symbol instead of the "=" symbol. We are giving an estimate, not an exact value.

ln(51) ≈ 3.9318

### Change of Base Rule

If a > 0, a ≠ 1, b > 0, b ≠ 1, and x > 0, then: $$\text{log}_{a}(x)=\frac{\text{log}_{b}(x)}{\text{log}_{b}(a)}$$ In the case where we need to approximate the value of a logarithm and the base is not 10 or e, we can use the change of base rule. This rule allows us to change the base of the logarithm. We can change it into a base 10 logarithm or a base e logarithm (natural logarithm) and then approximate its value using a calculator.Let's look at a few examples.

Example 3: Use the change of base rule and a calculator to approximate the answer (round to the nearest ten-thousandth). $$\text{log}_{3}(18)$$ Let's use the change of base rule. We can change the base to a base 10 log or a base e log (natural log). We will get the same answer either way. $$\text{log}_{3}(18)=\frac{\text{log}(18)}{\text{log}(3)}≈ 2.6309$$ If we used the natural log, we obtain the same answer: $$\text{log}_{3}(18)=\frac{\text{ln}(18)}{\text{ln}(3)}≈ 2.6309$$ Example 4: Use the change of base rule and a calculator to approximate the answer (round to the nearest ten-thousandth). $$\text{log}_{7}(\sqrt{5})$$ Let's use the change of base rule. We can change the base to a base 10 log or a base e log (natural log). We will get the same answer either way. $$\text{log}_{7}(\sqrt{5})=\frac{\text{log}(\sqrt{5})}{\text{log}(7)}≈ 0.4135$$ $$\text{log}_{7}(\sqrt{5})=\frac{\text{ln}(\sqrt{5})}{\text{ln}(7)}≈ 0.4135$$

### Deriving the Change of Base Rule

At this point, we know that logarithmic functions are one-to-one (pass the horizontal line test). If all variables are positive real numbers and we let x = y, then we can state: $$\text{log}_b(x)=\text{log}_b(y)$$ To start, let's let m be equal to log_{a}(x): $$\text{log}_a(x)=m$$ Write in exponential form: $$a^m=x$$ Take the logarithm with a base of b on each side: $$\text{log}_b(a^m)=\text{log}_b(x)$$ Use the power rule for logarithms: $$m \cdot \text{log}_b(a)=\text{log}_b(x)$$ Plug in for m: $$\text{log}_a(x) \cdot \text{log}_b(a)=\text{log}_b(x)$$ Divide both sides by log

_{b}(a): $$\text{log}_a(x)=\frac{\text{log}_b(x)}{\text{log}_b(a)}$$

#### Skills Check:

Example #1

Approximate each to the nearest thousandth. $$\text{log}_{4}(15)$$

Please choose the best answer.

A

$$2.698$$

B

$$1.953$$

C

$$2.122$$

D

$$1.753$$

E

$$1.113$$

Example #2

Approximate each to the nearest thousandth. $$\text{log}_{2}(6)$$

Please choose the best answer.

A

$$3.344$$

B

$$1.388$$

C

$$2.671$$

D

$$2.585$$

E

$$3.521$$

Example #3

Approximate each to the nearest thousandth. $$\text{log}_5(\sqrt{2})$$

Please choose the best answer.

A

$$0.579$$

B

$$0.702$$

C

$$0.215$$

D

$$0.46$$

E

$$0.953$$

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