How To Solve Logarithmic Equations Video
What is the general strategy for solving log equations?
Answer: As the video above points out, there are two main types of logarithmic equations. Before you to decide how to solve an equation, you must determine whether the equation
- A) has a logarithm on one side and a number on the other
- B) whether it has logarithms on both sides
Example 1 Logarithm on one side and a number on the other
General method for solving this type (log on one side), Rewrite the logarithm as exponential equation and solve. Let's look at a specific example:$$ log_4 x + log_4 8 = 3 $$
Step 1 Rewrite log side as single logarithm
$$ log_4 8x = 3 $$
$$ 4^ 3 = 8x $$
64 = 8x
8 = x
Example 2 Logarithm on both sides
General method to solve this kind (logarithm on both sides),Step 1 use the rules of logarithms to rewrite the left side and the right side of the equation to a single logarithm
Step 2 "cancel" the log
Step 3 solve the expression
Let's look at a specific ex $$ log_5 x + log_2 3 = log_5 6 $$
Step 1 rewrite both sides as single logs
$$ log_5 x + log_5 2 = log_5 6 \\ log_5 2x = log_5 6 $$
Step 2 "cancel" logs
$$ \color{Red}{ \cancel {log_5}} 2x = \color{Red}{ \cancel {log_5}} 6 \\ 2x = 6 $$
Step 2 Solve expression
x = 3
Practice Problems
Follow the steps for solving logarithmic equations with logs on both sides
rewrite both sides as single logs
$$ log_3 5x = log_3 15 $$
"cancel" logs
$$ \color{Red}{ \cancel{log_3}} 5x = \color{Red}{ \cancel{log_3}} 15 \\ 5x=15 $$
Solve expression
x = 3
Follow the steps for solving logarithmic equations with a log on one side
Rewrite log side as single logarithm
$$ log_3 9x = 4$$
$$ 3^4 = 9x $$
81 = 9 x
9 = x
Follow the steps on how to solve equations with logs on both sides
rewrite both sides as single logs<
$ log_3 5^2 + log_3 x = log_3 5^3 \\ log_3 25 +log_3 x = log_3 125 \\ log_3 25x = log_3 125 $
"cancel" logs
$ \color{Red}{ \cancel{log_3}} 25x = \color{Red}{ \cancel{log_3}} 125 \\ 25x = 125 $
Solve expression
x = 3
Follow the steps for solving logarithmic equations with a log on one side
Rewrite log side as single logarithm
$ 2 log_2 4 + log_2 x = 5 \\ log_2 4^2 = log_2 x = 5 \\ log_2 16 + log_2 x = 5 \\ log_2 16x = 5 $
25 = 16x
32 =16x
2 = x
Follow the steps on how to solve equations with logs on both sides
rewrite both sides as single logs
$ log_3 5^2 + log_3 x + log_3 5^3 \\ log_3 25x + log_3 125 $
log325x = log353
"cancel" logs
$ \color{Red}{ \cancel{log_3}} 25x + \color{Red}{ \cancel{log_3}} 125 \\ 25x=125 $
Solve expression
$ \frac{25x}{25} = \frac{125}{25} \\ $
Follow the steps on how to solve equations with logs on both sides
rewrite both sides as single logs
$ 2 log_3 7 - log_3 2x = log_3 98 \\ log_3 7^2 - log_3 2x = log_3 98 \\ log_3 49 - log_3 2x = log_3 98 \\ log_3 \frac{49}{2x} = log_3 98 $
"cancel" logs
$ \color{Red}{ \cancel{log_3}} \frac{49}{2x} = \color{Red}{ \cancel{log_3}} 98 \\ \frac{49}{2x} = 98 $
Solve expression
$ 49 = 196x \\ \frac{49}{196} = x \\ x = 49 $
You know the deal. Just follow the steps for solving logarithmic equations with logs on both sides
rewrite as single logs
$ 2 log_11 5 + log_11 x + log_11 2 = log_11 150 \\ log_11 5^2 + log_11 2x = log_11 150 \\ log_11 25 + log_11 2x = log_11 150 \\ log_11 50x= log_11 150 $
2log115 + log11x+ log112 = log11150
"cancel" logs
$ \color{Red}{ \cancel{log_1}} 50x = \color{Red}{ \cancel{log_11}} 150 \\ 50x = 150 $
Solve expression
x = 3
