Question

22 Solve the following equations:
(1) log (2x + 3) = log 7
(ta) log + 1) + lng (​

Answer 1

Answer:

When solving logarithmic problems, there are several different types of logarithmic problems we could be

asked to solve. For example, we could be asked to solve any of the following:

2

log (x 5) 3 + = 9 9 log (5x 2) log (3x 8) + = + log(x 1) log(x 2) 2 + + − =

The three examples shown above all require slightly different strategies to solve. The notes “Solving

Logarithmic Equations – Part 1”, focused on solving problems similar to the first example. That is,

logarithmic problems that contain only one logarithm. Now we are going to do a quick review of solving

logarithmic problems containing only one logarithm, then look at solving the other two types.

Solving Logarithmic Equations Containing Only One Logarithm

Remember the key to solving logarithmic problems that only contain one logarithm is to rewrite the problem

in exponential form. After we have rewritten the logarithmic problem in exponential form, we should then

be able to solve the resulting equation. Let’s work through an example to refresh our memory.

Example 1: Solve

2

7

log (2x 3x 1) 0 + − = 2 2 0

7

log (2x 3x 1) 0 2x 3x 1 7 + − = → + − =

Rewrite the problem in exponential form.

2 0

2

2

2x 3x 1 7

2x 3x 1 1

2x 3x 2 0

(2x 1)(x 2) 0

1

x , 2

2

+ − =

+ − =

+ − =

− + =

= −

When you plug

1

x

2

=

or x = –2 into 2x2 + 3x – 1 you get 1, a positive number.

Example 2: Solve

ln(4x 3) 2 + = 2

ln(4x 3) 2 4x 3 e + = → + =

Rewrite the problem in exponential form.

4x 3 7.389056

4x 4.389056

x 1.097264

+ 





When you plug x ≈ 1.097264 into (4x + 3) you get a positive number.

Solving Logarithmic Equations Containing Only Logarithms

Let’s consider the following problem from above,

9 9 log (5x 2) log (3x 8). + = +

After observing that the

logarithmic equation contains only logarithms, what is the next step? Consider the following:

This statement says that if an equation contains only two logarithms, on opposite sides of the equal sign,

with the same base then the problem can be solved by simply dropping the logarithms. Now let’s work

through a few examples starting with the problem from above.