In these lessons, we will look at how to evaluate simple logarithmic functions and solve for x in logarithmic functions.
More Logarithms and Algebra Lessons
Solving for x in Logarithmic Equations
Equations of the form x
= loga y
can be solved (for any of the three variables y, a
) by first writing them in exponent form. We must be careful to check the answer(s) to see whether the logarithm is defined.
Calculate the value of each of the following:
a) 1og 2 64
b) log 9 3
c) log 4 1
d) log 6 6
e) log 8 0.25
f) log 3–9
a) Let x = log 2 64
2x = 64
x = 6
b) Let x = log 9 3
9x = 3
c) Let x = log 4 1
4x = 1
x = 0
d) Let x = log 6 6
6x = 6
x = 1
e) Let x = log 8 0.25
8x = 0.25
f) Let x = log 3– 9
3x = – 9
Since it is not possible for 3x
to be negative, log 3
–9 is undefined.
Solve logx 4 = 2
logx 4 = 2
x2 = 4
x = 2 or –2
Since x is the base, x > 0 and x ≠ 1; so x = –2 is rejected and the only solution is x = 2
Solve log 3 x = 2
log 3 x = 2
32 = x
x = 9
Solve log x (4x – 3) = 2
log x (4x – 3) = 2
x2 = 4x – 3
x2 – 4x + 3 = 0
(x -1)(x – 3) = 0
So, x = 1 or 3
For the logarithm to be defined, the only solution is 3.
Take note of the following:
- Logarithms of a number to the base of the same number is 1, i.e. loga a = 1
- Logarithms of 1 to any base is 0, i.e. loga 1 = 0
- Loga 0 is undefined
- Logarithms of negative numbers are undefined.
- The base of logarithms cannot be negative or 1.
Solving Logarithmic Equations
Just as we can use logarithms to access exponents in exponential equations, we can use exponentiation to access the insides of a logarithm. Solving logarithmic equations often involves exponentiating logarithms in order to get rid of the log and access its insides. Sometimes we can use the product rule, the quotient rule, or the power rule of logarithms to help us with solving logarithmic equations.
This video shows how solve a logarithmic equation using properties of logarithms and some other algebra techniques.
Solving a Logarithmic Equation with Multiple Logs
When given a problem on solving a logarithmic equation with multiple logs, students should understand how to condense logarithms. By condensing the logarithms, we can create an equation with only one log, and can use methods of exponentiation for solving a logarithmic equation with multiple logs. This requires knowledge of the product, quotient and power rules of logarithms.
Have a look at the following video for more examples on solving logarithms.:
Techniques for Solving Logarithmic Equations.
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