Chapter 12 Formulas and Hints
I) Definitions
Exponential Equation: An equation that does NOT have logarithms.
Example: 4 x  7 Where: 4 is the base, "x” is the exponent, and 7 is the target.
(Note: The exponential equation above is written as a logarithmic equation below.)
Logarithmic Equation: An equation that has logarithms.
Example: x  log4 7 Where: 4 is the base, “x” is the exponent, and 7 is the target.
Base 10 Logarithms: A logarithmic equation that does NOT have a base shown is base 10. Base
10 logarithms can be computed using most calculators.
Example: x  log 16 => x  1.2041
Base “e” Equations: An exponential equation with base “e” (2.718281828).
Example: 7  e x
“Natural” Logarithms: A logarithmic equation with base “e”. Because natural logarithms
occur so frequently, they are shown as “ln” on most calculators.
Example: x  ln 12 is the same as x  loge 12
II) Logarithm Properties (Applicable only to the “Target” of the logarithm)
Product / sum:
log a ( x  y)  log a x  log a y
Example: log c [ x   x  6 ]  logc x  logc  x  6
Quotient / difference:
 x
log a    log a x  loga y
 y
z
Example: log q    log q z  logq 3
3
Exponent / product: loga xP  P  loga x
Example: logs z 7  7  log x z
III Solving Equations
Non-Base “10” Conversions: Use the following conversion formula loga t 
Example: log 4 18 
log10 18
log18
 2.08 or log 4 18 
 2.08
log10 4
log 4
log10 t
log10 a
Exponential Equations: To solve an exponential equation 1) try to write it as a logarithmic
equation. If exponents are on both sides of the equation, 2) take the log of both sides.
Example: 1) 13  10x => x  log10 13 => x  1.1139
2) 3x  10x2 => log 3x  log10 x2 => x  log 3   x - 2   log 10
 0.477 x  1 x  2  => 0.477 x  x  2 =>  0.523x  2 => x  3.824
Logarithmic Equations: To solve a logarithmic equation:
1)
2)
3)
4)
Combine multiple logarithms into a single logarithm using Logarithmic Properties.
Write the logarithmic equation as an exponential equation.
Solve
Check your answers. Remember a logarithm’s target must be > 0.
Example: 2  log 3 x  log 3 x  4
1) The target of a logarithm must be greater than 0, so x > 0.
2)
3)
4)
 x2 
2  log3 x  log3 x  4 => log3 x 2  log3 x  4 => log3    4 => log3 x  4
 x
4
log3 x  4 => 3  x
x  81 Is x > 0? Yes. So the solution is valid.
III Miscellaneous
Finding Inverses of Functions: To find the inverse of a function:
1) Note any values for “x” that will NOT result in real numbers.
2) Write “f(x)” as “y”. 3) Swap the “x” and “y”. 4) Solve for “y”.
5) Write “y” as “f -1(x)”
Example: f  x   x  2
1) Any negative values under a radical result in imaginary numbers. So x  2 !
2) y  x  2
3) x  y  2 4) x  y  2 => x 2  y  2 => y  x 2  2
5)
Note:
f 1  x   x 2  2 Where x  2 .
f 1  f  a    a and f  f 1  a    a ; Examples: f 1  f  23   23 and f  f 1 102    102
Simplification of Logarithms:
aloga x  x and log a a x  x ;
log518
Examples: 5
 18 and log3 35  5
Graphing Exponential and Logarithmic Equations: To graph both an exponential and logarithmic
equation, enter the equation into your calculator and then look at table values. Select points from the
table which can be graphed. (Note: To get points for a logarithmic graph I recommend using the
base “10” conversion formula and plugging the resultant equation into your calculator.)