Logarithm Problems?

Question

Solve exactly and approximately to 3 decimal places.

1. log 3 (x+5) + log 3 (x-3) = 2

Log 3 is log base of 3.

My answer: x = 4

2. 3^(5x-1) = 81

My answer: x = 1

3. 2log 2 (3x+2) - log 2 (x) = 6

Log 2 is log base of 2.

My answer: x = 2^(3x-4)

4. 3^(4x-3) = 100

My answer: x = (ln 100 + ln 3 (3) ) / ln 3 (4)
x = 1.7980

5. log 5 (x+8) + log 5 (x+4) = 1

log 5 is log base of 5

My answer: x = - 3


Am I right for these? Thanks.

Answer 1

1)

log3(x+5) + log3(x-3) = 2

log3(x+5)(x-3) = 2 (since log a + log b =logab)

(x+5)(x-3) = 3^2 = 9

(x^2 + 2x - 15) = 9

x^2 + 2x -24 = 0

(x +6)(x-4) = 0

x = -6 or 4

2)

3^(5x-1) = 81

3^(5x-1) = 3^4

5x-1 = 4

5x = 5

x = 1

3)

2 log2(3x+2) - log2(x) = 6

log2(3x+2)^2 - log2(x) = 6

log2(3x+2)^2/(x)=6

(3x+2)^2/x = 2^6

(9x^2 + 12x + 4)/x = 64

(9x^2 + 12x + 4) = 64x

9x^2 - 52x + 4 = 0

x = 52+/- sqrt(2704 - 144)/18

x = 42+/- sqrt(2560)/18

x = 42+/-sqrt(16^2*10)/18

x = 42+/-16sqrt(10)/18

x = (1/9)[21+/-8 sqrt(10)

4)

3^(4x-3) = 100

take logs to the base 10

(4x-3) log(3) = log100

(4x-3)log(3) = 2(since log 100 to the base 10 =2)

(4x-3) = 2/log(3)

4x-3 = 2/0.477

4x -3 = 4.193

4x = 7.193

x = 7.193/4

x = 1.798

5)
log5(x+8) + log5(x+4) = 1

log5(x+8)(x+4) = 1

5^1 = (x+8)(x+4)

x^2 + 12x + 32 = 5

x^2 + 12x + 27 = 0

(x+3)(x+9)=0

x = -3 or -9

Answer 2

(1)
log[b] x = ln(x)/ln(b)

ln(x + 5) + ln(x - 3) = 2 ln(3)
ln((x+5)(x-3)) = 2 ln(3)
exp(ln(x+5)(x-3)) = exp(2ln(3))
(x+5)(x-3) = exp(ln(9)) = 9
x² + 2x - 15 = 9
x² + 2x - 24 = 0

x = 4, -6

reject the negative answer, the log of a negative number is a complex number.

1 is correct.

3 is not right, see if you get 26/9 + (8/9)sqrt(10) using the same method that I used above.

Answer 3

First of all, thank you for posting your answers!

#1 is correct.

#2 is correct.

#3: you need to solve for x. Recall that a*log (w) = log (w^a) so convert the entire left side to log of something. Then, as usual, exponentiate both sides (with 2 as the base here) to get rid of the log. There are two solutions. (I get about 5.7 and 0.078)

#4 is correct.

#5 is correct.

Good job!