Solve the exponential equation using natural logs

Question

2^x-3 = 7

Answer 1

Hi,

2^x-3 = 7

Take the natural log (ln) of each side.

ln(2^x-3) = ln(7)

Remember the power rule for logs: log_ba^r = rlog_ba.

So, (x-3)ln(2) = ln(7).

Divide by ln(2) on each side: x - 3 = ln(7)/ln(2)

Add 3 to each side. x = 3 + ln(7)/ln(2). Use your calculator to solve! x = 3 + (1.946/0.693) = 3 + 2.808 = 5.808.

Answer 2

2^(x-3) = 7

x-3 = slightly more than 2.8, as 2^2.8 = about 7

if the base is the same, then the powers are the same

2^(x-3) = 2^2.8, then

x-3 = 2.8 and

x = about 5.8

2^(x-3) = 7

take natural logs of both sides:

ln(2^(x-3)) = ln7

the power comes out as a coefficient:

(x-3)ln2 = ln7

divide both sides by ln2

x-3 = ln7/ln2

add 3 to both sides

x = 3 + ln7/ln2

use a calculator with a natural log function

x = about 3+1.9459/.6931

= about 3+ 2.80734

= about 5.80734

Answer 3