exponential equations

Question

First: Solve each exponential equation. Give answer in logarithmic form, then estimate. Round all values to 3 decimal places where needed. 

1) 2(10^3x) = 48

2)-2-4^2x=-13

--------------------------------------------

Second:

use properties of logs to solve each logarithmic equation. Round to 4 decimal places where needed. 

1) -8+2log₃(3x+5)=2

Answer 1

1)

 

Divide both sides of the equation 2.

 

(10)^3x = 24

 

 

Log both sides of the equation and bring down any exponent as the coefficient of the log.

 

3xLog(10) = Log(24)

 

 

Note that Log(10) is 1.

 

3x = log(24)

 

 

2)

 

Add 2 to both sides of the equation.

 

4^2x = -11

 

 

Log both sides of the equation and bring down any exponent as the coefficient of the log.

 

2xLog(4) = Log(-11)

 

This one has no solution, since the argument of a log cannot be negative.

 

 

SECOND:

 

1)

 

We can rewrite -8 as -log₃(6561).

 

-log₃(6561) + log(3x + 5)² = 2

 

Log₃[(3x + 5)² / 6561] = Log₃(9)

 

 

Equate arguments.

 

(3x + 5)² / 6561 = 9

 

 

Now you can solve for x. I leave this part to you.