y = logb (x) If and only if: by = x Note that b is the base of the logarithm. It must also be true that: b > 0 b does not equal 1 In the same equation, y is the exponent and x is the exponential expression that the logarithm is set equal to.
If and only if: by = x
Example: 5 = log4(1024) b = 4 y = 5 x = 1024
b = 4 y = 5 x = 1024
Example: 1024 = ?
Example: 4 * 4 * 4 * 4 * 4 = ? This could also be written as: 45
This could also be written as: 45
Example: 45 = 1024
Example: log3(x + 5) + 6 = 10 log3(x + 5) + 6 - 6 = 10 - 6 log3(x + 5) = 4
log3(x + 5) + 6 - 6 = 10 - 6 log3(x + 5) = 4
Example:log3(x + 5) = 4 Comparing this equation to the definition [y = logb (x)], you can conclude that: y = 4; b = 3; x = x + 5 Rewrite the equation so that: by = x 34 = x + 5
Comparing this equation to the definition [y = logb (x)], you can conclude that: y = 4; b = 3; x = x + 5 Rewrite the equation so that: by = x 34 = x + 5
Example: 34 = x + 5 3 * 3 * 3 * 3 = x + 5 81 = x + 5 81 - 5 = x + 5 - 5 76 = x
3 * 3 * 3 * 3 = x + 5 81 = x + 5 81 - 5 = x + 5 - 5 76 = x
Example: x = 76
logb(m * n) = logb(m) + logb(n) Also note that the following must be true: m > 0 n > 0
Example: log4(x + 6) = 2 - log4(x) log4(x + 6) + log4(x) = 2 - log4(x) + log4(x) log4(x + 6) + log4(x) = 2
log4(x + 6) + log4(x) = 2 - log4(x) + log4(x) log4(x + 6) + log4(x) = 2
Example: log4(x + 6) + log4(x) = 2 log4[(x + 6) * x] = 2 log4(x2 + 6x) = 2
log4[(x + 6) * x] = 2 log4(x2 + 6x) = 2
Example: log4(x2 + 6x) = 2 Comparing this equation to the definition [y = logb (x)], you can conclude that: y = 2; b = 4 ; x = x2 + 6x Rewrite the equation so that: by = x 42 = x2 + 6x
Comparing this equation to the definition [y = logb (x)], you can conclude that: y = 2; b = 4 ; x = x2 + 6x Rewrite the equation so that: by = x 42 = x2 + 6x
Example: 42 = x2 + 6x 4 * 4 = x2 + 6x 16 = x2 + 6x 16 - 16 = x2 + 6x - 16 0 = x2 + 6x - 16 0 = (x - 2) * (x + 8) x = 2; x = -8
4 * 4 = x2 + 6x 16 = x2 + 6x 16 - 16 = x2 + 6x - 16 0 = x2 + 6x - 16 0 = (x - 2) * (x + 8) x = 2; x = -8
Example: x = 2 Note that you cannot have a negative solution for a logarithm, so you can discard x - 8 as a solution.
logb(m / n) = logb(m) - logb(n) Also note that the following must be true: m > 0 n > 0
Example: log3(x + 6) = 2 + log3(x - 2) log3(x + 6) - log3(x - 2) = 2 + log3(x - 2) - log3(x - 2) log3(x + 6) - log3(x - 2) = 2
log3(x + 6) - log3(x - 2) = 2 + log3(x - 2) - log3(x - 2) log3(x + 6) - log3(x - 2) = 2
Example: log3(x + 6) - log3(x - 2) = 2 log3[(x + 6) / (x - 2)] = 2
log3[(x + 6) / (x - 2)] = 2
Example: log3[(x + 6) / (x - 2)] = 2 Comparing this equation to the definition [y = logb (x)], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2) Rewrite the equation so that: by = x 32 = (x + 6) / (x - 2)
Comparing this equation to the definition [y = logb (x)], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2) Rewrite the equation so that: by = x 32 = (x + 6) / (x - 2)
Example: 32 = (x + 6) / (x - 2) 3 * 3 = (x + 6) / (x - 2) 9 = (x + 6) / (x - 2) 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2) 9x - 18 = x + 6 9x - x - 18 + 18 = x - x + 6 + 18 8x = 24 8x / 8 = 24 / 8 x = 3
3 * 3 = (x + 6) / (x - 2) 9 = (x + 6) / (x - 2) 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2) 9x - 18 = x + 6 9x - x - 18 + 18 = x - x + 6 + 18 8x = 24 8x / 8 = 24 / 8 x = 3
Example: x = 3
