solving exponential equations equations with variables in exponents, such as 3 x = 5 and 7 3x = 90...
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Solving Exponential Equations
Equations with variables in exponents, such as
3x = 5 and 73x = 90
are called exponential equations.
In Section 9.3, we solved certain logarithmic equations by using the principle
am = x means logax = m
Solution
Example Solve: 3 x +1 = 43
We have 3 x +1 = 43
log 3 x +1 = log 43
(x +1)log 3 = log 43
x 2.424.
Principle of logarithmic equality
Power rule for logs
log 431
log3x
log 431
log3x Shuhaw Answer
Chemistry Answer
Solve graphically
x 2.424
Solution
Example Solve: e1.32t = 2000
We have:
t 5.758.
Note that we use the natural logarithm
Logarithmic and exponential functions are inverses of each other
e1.32t = 2000
ln e1.32t = ln 2000
1.32t = ln 2000
ln 2000
1.32t
To Solve an Equation of the Form at = b for t1. Take the logarithm (either natural or common) of both sides.
2. Use the power rule for exponents so that the variable is no longer written as an exponent.
3. Divide both sides by the coefficient of the variable to isolate the variable.
4. If appropriate, use a calculator to find an approximate solution in decimal form.
ln lnta b log logta b
ln lnt a b log logt a b
ln
ln
bt
a
log
log
bt
a
Solution
Example Solve: log2(6x + 5) = 4.
6x + 5 = 24
6x = 11
The solution is x = 11/6.
log2(6x + 5) = 4
6x + 5 = 16
x = 11/6
logax = m means am = x
Solving Logarithmic Equations
Solve graphically
x = 11/6
Using change of base 1ln(6 5)
(4)ln(2)
xY
Solution
Example Solve: log x + log (x + 9) = 1.
x2 + 9x = 10
To increase the understanding, we write in the base 10.
log10 x + log10 (x + 9) = 1
log10[x(x + 9)] = 1
x(x + 9) = 101
x2 + 9x – 10 = 0
(x – 1)(x + 10) = 0
x – 1 = 0 or x + 10 = 0 x = 1 or x = –10
0 + log (10) = 1
x = 1:
log 1 + log (1 + 9) = 1
0 + 1 = 1 TRUE
x = –10:
log (–10) + log (–10 + 9) = 1 FALSE
The solution is x = 1.
The logarithm of a negative number is undefined.
Solve graphically
We graph y1= log(x) + log (x + 9) - (1)
x = 1
Solution
Example Solve: log3(2x + 3) – log3(x – 1) = 2.
log3(2x + 3) – log3(x – 1) = 2
(2x + 3) = 9(x – 1)
x = 12/7
2x + 3 = 9x – 9
The solution is 12/7. Check is left to the student.
32 3
log 21
x
x
22 33
1
x
x
2 39
1
x
x
Solution
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