Solve the following equations, giving exact values if possible.

Question

2. Solve the following equations, giving exact values if possible.
a. 9^x-1 = (1/3)^2x
b. ln(x+2)=3
c. log27(x)=1-log27(x-0.4)
d. logb6=1/2

Answer 1

1)

 

9^{x + 1} = (1/3)^2x

 

 

Convert to common base.

 

(3²)^{x - 1} = (3^-1)^2x

 

 

Equate exponents.

 

2(x - 1) = -2x

 

 

Solve for x.

 

 

2)

 

ln(x + 2) = 3

 

 

The solution to a natural log is the exponent of base number e.

 

e³ = x + 2

 

 

Solve for x.

 

 

3)

 

Move all log to one side of the equation, and non-log terms to the other side.

 

log₂₇x + log₂₇(x - 0.4) = 1

 

 

Condense the logs using the properties.

 

log₂₇(x(x - 0.4)) = 1

 

log₂₇(x² - 0.4x) = 1

 

 

1 is the same as log₂₇(27).

 

log₂₇(x² - 0.4x) = log₂₇(27)

 

 

Equate the arguments of logs.

 

x² - 0.4x = 27

 

Solve for x.

 

 

4)

 

log_b6 = 1/2

 

b^1/2 = 6

 

√b = 6

 

Solve for b.