Logarithmic Equations: Introduction and Simple Equations
This discussion will focus on the common logarithmic functions.
The general common logarithmic equation is:
COMMON LOGARITHMIC FUNCTION
$y=lo{g}_{a}x$ if and only if x = a^{y}
Where a > 0, a ≠ 1 and x > 0
When reading $lo{g}_{a}x$ say, "log base a of x".
Some examples are:
1. $lo{g}_{10}100=2$ because 10^{2} = 100
2. $lo{g}_{3}81=4$ because 3^{4} = 81
3. $lo{g}_{15}225=2$ because 15^{2} = 225
Notice in the examples that the base of the log is also the base of the corresponding exponent. In example 1 above, the logarithmic function has a log of base 10 and the corresponding exponential function has a base of 10.
If you see log with no base is means log of base 10 or log = log_{10}.
Some basic properties of logarithmic functions are:
Property 2: $lo{g}_{a}a=1$ because a^{1} = a
Property 3: If $lo{g}_{a}x=lo{g}_{a}y$, then x = y One-to One Property
Property 4: $lo{g}_{a}{a}^{x}=x$ and ${a}^{{\text{log}}_{a}x}=x$ Inverse Property
Let's solve some simple logarithmic equations:
Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate. |
Property 4 - Inverse |
Step 2: Apply the Property. Remember $log=lo{g}_{10}$. Since the log has a base of 10, taking the inverse means to rewrite both sides as exponents with base 10. |
log x = 4 Original 10^{logx} = 10^{4} Exponent of 10 |
Step 3: Solve for x. Property 4 states that ${a}^{lo{g}_{a}x}=x$, therefore the left-hand side becomes x. |
x = 10^{4} Apply Property x = 10,000 Evaluate |
Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Since a log is set equal to a log of the same base. Property 3 is the most appropriate. |
Property 3 - One to One |
Step 2: Apply the Property. Property 3 states that if $lo{g}_{a}x=lo{g}_{a}y$, then x = y. Therefore x = 4x - 9. |
x = 4x - 9 Apply Property |
Step 3: Solve for x. |
-3x = -9 Subtract 4x x = 3 Divide by -3 |
Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the log equals neither 0 nor 1. Property 3 does not apply since a log is not set equal to a log of the same base. Therefore Property 4 is the most appropriate. |
Property 4 - Inverse |
Step 2: Apply the Property. Since the log has a base of 3, taking the inverse means to rewrite both sides as exponents with base 3. |
$lo{g}_{3}3x=5$ Original ${3}^{{\mathrm{log}}_{3}3x}={3}^{5}$ Exponent of 3 |
Step 3: Solve for x. Property 4 states that ${a}^{lo{g}_{a}x}=x$, therefore the left-hand side becomes x. |
3^{x} = 3^{5} Apply Property $x=\frac{243}{3}$ Divide by 3 x = 81 Evaluate |
Related Links: Math algebra Logarithmic Equations: Natural Base - Simple Equations Algebra Topics Logarithmic Equations |
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