Algebra

##### Related Lessons

- Adding and Subtracting Scientific Notation
- Adding Fractions
- Algebra Calculators
- Arithmetic Sequence
- Average Rate of Change
- Change of Base Formula
- Cm to M
- Commutative Property of Addition
- Completing the Square
- Cross Product Calculator
- Determinant Calculator
- Determinant of a Matrix
- Difference of Squares
- Discriminant
- Divisibility Rule for 7
- Dot Product Calculator
- Eigenvalue Calculator
- Eigenvector Calculator
- Equation of a Circle
- Even Numbers
- Exponent Rules
- Factorial Calculator
- Fractional Exponents
- How to Find the Median
- Interval Notation
- Matrix Addition
- Matrix Subtraction
- Midpoint Formula
- Multiplying Negative Numbers
- Negative Exponents
- Odd Numbers
- One to One Function
- Partial Fraction Decomposition Calculator
- Point Slope Form
- Properties of Multiplication
- Rationalize the Denominator
- Rectangular to Polar Calculator
- Reflexive Property
- Round to the Nearest Tenth
- RREF Calculator
- Slope Calculator
- Slope Intercept Form
- Standard Form
- Summation Calculator
- Vertex Form

Tutors/teachers:

Nikkolas

Tutor and Aerospace Engineer

# Change of Base Formula

## The Change of Base Formula

The change of base formula is given as:

Where *b* is the original base of the logarithm, and *d* is the base that we are changing to.

### Why do we Perform a Change of Base?

**Sometimes we are faced with a logarithm that we are unable to evaluate given its current base.** For example, we may have to calculate log_{6}(3) but our calculator can only perform base 10 logarithms. This is where the change of base formula can come in and save the day.

Another situation where it proves to be extremely useful is when trying to solve logarithms by hand. There are certain combinations of logarithm base and logarithm subject that can be solved without the use of a calculator. The change of base formula can help us format a logarithm into one of these combinations.

### Rules to Follow

Using the change of base formula allows us to calculate a logarithm of any base b, with restrictions that b > 0 and b ≠ 1. These restrictions are in place because if b ≤ 0 or b = 1, the result will be indeterminate (meaning we will be unable to get the answer).

### Example Problem

Evaluate log_{4}(12) as if the calculator is only able to compute logarithms of base e (Euler’s number).

Solution:

Let’s set b = 4, x = 12, and d = e. Using e as our logarithm base for evaluation allows us to use the natural logarithm, notated as *ln*.

Plugging our constants into the change of base formula gives us:**The logarithm log _{4}(12) evaluated as ^{ln(12)}/_{ln(4)} results in 1.7925.**

Result :

Worksheet 1

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Cheat sheet

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