# Logarithms Review and Practice Questions

- Posted by Brian Stocker MA
- Date June 20, 2014
- Comments 1 comment

A Logarithm is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

Quick Review — Practice Questions — Answer Key — Common Mistakes Answering Logarithm

## Logarithms – a Quick Review

Logarithm is a function that has the form

*log _{y} x = a*

It actually solves this equation: which number do we put as a degree on the variable *y* to get the variable *x, *that is:

*y ^{a }= x*

*y*is called the base and

*a*is the exponent.

For example, let’s solve logarithm *log _{5}25 = a.*

Here, we represent 25 using 5 and the second degree. *a* and 2 are both on the number 5, so they must be the same.

* 5 ^{a }= 25*

*5*

^{a }= 5^{2}Here, we represent 25 using 5 and the second degree. *a* and 2 are both on the number 5, so they must be the same.

* a = 2*

We can see from the way the logarithm works, that:

*log _{a}1 = 0* and

*log*

_{a}a = 1From *log _{a}1 = 0* we have that

*a*, which is true for any real number

^{0 }= 1*a*.

From

*log*we have that

_{a}a = 1*a*, which is true for any real number

^{1}= a*a*.

If in the logarithm the base is 10, then instead of* log _{10}* we write

*lg.*

When we are solving some logarithm, any part can be unknown. In the first example, we had a case where the exponent was the unknown variable. Let’s see another example, where both exponent and base are known:

*lg x = 2*

*10 ^{2 }= x*

*x = 100*

## Logarithm Practice Questions

**1. If log_{2}x = 3, then x is:**

A. 9

B. 8

C. 7

D. 6

**2. Solve the equation log_{4}1/4 = x.**

A. -1

B. 0

C. 1

D. 2

**3. For what x is the following equation correct:**

** log_{x}125 = 3**

A. 1

B. 2

C. 3

D. 5

**4. Find x if log_{x}(9/25) = 2.**

A. 3/5

B. 5/3

C. 6/5

D. 5/6

**5. Solve log_{10}10,000 = x.**

A. 2

B. 4

C. 3

D. 6

**6. Find x if log_{1/2 }x = 4.**

A.16

B. 8

C. 1/8

D. 1/16

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## Answer Key

**1. B. 8**

*log _{2}x = 3
*

*2*

^{3 }= x*x = 8*

**2. A. -1**

*log _{4}1/4 = x.
*

*4*

^{x }= 1/4*4*

^{x }= 4^{-1 }*x = -1*

**3. E. 5**

*log _{x}125 = 3
*

*x*

^{3 }= 125*x*

^{3 }= 5^{3 }*x = 5*

**4. A. 3/5**

*log _{x}(9/25) = 2
*

*x*

^{2 }= 9/25*x*

^{2 }= (3/5)^{2 }*x = 3/5*

**5. B. 4**

*log _{10}10,000 = x
*

*10*

^{x }= 10,000*10*

^{x }= 10^{4 }*x = 4*

**6. D. 1/16**

*log _{1/2 }x = 4
*

*(1/2)*

^{4 }= x*x = 1/16*

## Common Mistakes Answering Logarithm Questions

**Misunderstanding Logarithmic Properties**- Confusing the properties of logarithms, such as the product rule, quotient rule, and power rule.

**Converting the Base Incorrectly**- Make sure you keep the bases straight! Incorrectly converting different logarithmic bases will give an incorrect answer.

**Simplification Errors**- Making mistakes in simplifying logarithmic expressions.

**Logarithmic Equation Solving**- Be careful when the solution requires combining or separating logarithmic terms.

**Misinterpreting the Logarithm Function**- Remember the relationship between logarithms and exponentiation is inverse.

**Watch Domain Restrictions****Calculation Mistakes**- Check your work to avoid simple arithmetic or algebraic errors.

**Incorrectly applying Logarithmic Properties****Handling Complex Logarithmic Expressions**:- Be careful solving complicated expressions with multiple logarithms and require combining several properties.

**Remember the Logarithm of 1**- The logarithm of 1 is always 0, for any base.

**Natural Logarithms**- Confusing natural logarithms (ln) with logarithms of other bases.

**Remember and Apply Exponent Rules**- Apply exponent rules when converting between logarithmic and exponential forms.

**Incorrectly Applying Logarithmic Scale****Incorrectly Applying Change of Base Formula**:

**Written by**, Brian Stocker MA., Complete Test Preparation Inc.

**Date Published:**Friday, June 20th, 2014

**Date Modified:**Wednesday, June 5th, 2024

## 1 Comment

Thanks!