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5.6 LOGARITHMIC FUNCTIONS
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5.6 LOGARITHMIC FUNCTIONS
Inquiry
Why do we specify that the number you are taking the logarithm of has to be positive?
Any positive base raised to a power (even a negative power) yields a positive number!
5^(-2) = 1/ 5^2 = 1/25 or .04
So, 5^x = .04
x = log(5).04 use change of base formula
x = log(.04)/log(5) = -2
Exponents & Logarithms
VIDEOS
b, the base = 10
2^3 = 8
When you use the change of base formula, you are changing each base to base 10.
If y = 10^x, then x = log(y) (assumed base 10),
So 10 = 10^x, then log(10) = 1, 10^1 = 10
100 = 10^x, then log(100) = 2, 10^2 = 100
1000 = 10^x, then log(1000) = 3, 10^3 = 1000
The expression log(x) is another way of expressing x as an exponent on the base 10. Ten is the common base for logarithms, so log(x) is called a common logarithm.
Basic Rule
Since the inverse of an exponential function is a logarithmic function, then y = b^x has an inverse of
x = b^y. Now aply the log rule, y = log(b)x,
for a > 0 and b > 0.
4^3 = 64