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LOGS EQUAL THE
Logarithmic Function
The inverse of an exponential function is a
logarithmic function.
x = log a y
read: “x equals log base a of y”
We can convert exponential equations to
logarithmic equations and vice versa, using this:
Logarithmic form: y = logb x
Exponential Form: by = x.
The log to the base “b” of “x”
is the exponent to which “b”
must be raised to obtain “x”
y = log2 x

y = log b x 
x=2
y
x=by
Convert to exponential form
1)
ylog 5
3
2)
3)
2log a 7
alog bd
3 5
y
2
a 7
b d
a
Convert to logarithmic form
4)
5)
6)
10  1000
3
2 8
x
y
1
4
3log 1000
10
xlog 8
2
1log 4
y
Now that we can convert between the
two forms we can simplify logarithmic
expressions. Without a Calculator!
Simplify
“What is the exponent of that gives you 32?”
1) log2 32=x
2x = 32
x=5
“What is the exponent of 3 that gives you 27?”
2) log3 27=x
3x = 27
x=3
3) log4 2=x
4x = 2
x = 1/2
4) log3 1=x
3x = 1
x=0
Evaluate
1
5) log 6
x
36
We can also use these two forms
to help us solve for an inverse.
The steps for finding an inverse are the
same as before.
Easy as 1, 2, 3…
1-Rewrite
2-Switch x and y
3-Solve for y
Example: Find the inverse
y 8
x
Isolate the power.
1. rewrite (no-need)
2. Switch x and y
3. Solve for y
Rewrite in log form :

y  8x
x  8y
x  8y
x 8
log8 x  y
y
y  log8 x is the inverse of y  8
x
Find the inverse of the following
exponential functions…
f(x) = 2x
f(x) = 2x+1
f(x) = 3x- 1
f-1(x) = log2x
f-1(x) = log2x f-1(x) = log3(x + 1
Find the inverse of the log
functions.
y  log 2 x  4
1. Switch x and y
2. Isolate the log
3. Rewrite in
exponential form.
4. Solve for y.
Find the inverse for the following:
1. y = log3x
2. y = log(2x)
3. y = log5x – 3
Info about Logarithms
Common logarithms – have a base of
10, but we do not write the base
Ex: log10x = log x
Natural logarithms – have a base of e,
logex = ln x