How do you simplify log (x+6)=1-log(x-5)log(x+6)=1log(x5)?

2 Answers
May 11, 2018

I tried this:

Explanation:

We can use, as an example, the base of the logs as being 1010. We get:

log_(10)(x+6)=1-log_(10)(x-5)log10(x+6)=1log10(x5)

rearrange:

log_(10)(x+6)+log_(10)(x-5)=1log10(x+6)+log10(x5)=1

use a property of logs to write:

log_(10)[(x+6)(x-5)]=1log10[(x+6)(x5)]=1

use the definition of log to write:

(x+6)(x-5)=10^1(x+6)(x5)=101

that becomes:

x^2-5x+6x-30-10=0x25x+6x3010=0
x^2+x-40=0x2+x40=0

Use the Quadratic Formula:

x_(1,2)=(-1+-sqrt(1+160))/2=(-1+-sqrt(161))/2x1,2=1±1+1602=1±1612
and:
x_1=(-1+sqrt(161))/2=5.8442x1=1+1612=5.8442
x_2=(-1-sqrt(161))/2=-6.8442x2=11612=6.8442

the negative solution cannot be accepted.

May 11, 2018

x = (-1 + sqrt(161))/2 ~~ 5.844x=1+16125.844

Explanation:

log(x+6) = 1 - log(x-5)log(x+6)=1log(x5)

Move log(x-5)log(x5) to Left Side
log(x+6) + log(x-5) = 1log(x+6)+log(x5)=1

Condense the Left Side into 1 Log expression
log((x+6)(x-5)) = 1log((x+6)(x5))=1

Convert from Log Equation to Exponential Equation
(x+6)(x-5) = 10^1(x+6)(x5)=101

Multiply Left Side
x^2 - x - 30 = 10x2x30=10

Move 1010 to Left Side
x^2 - x - 40 = 0x2x40=0

Since we can't factor, use Quadratic Formula:
Quadratic Formula: x = (-b +- sqrt(b^2 - 4ac))/(2a)x=b±b24ac2a
x = (-1 + sqrt(161))/2 ~~ 5.844x=1+16125.844
And
x = (-1 - sqrt(161))/2 ~~ -6.844x=116126.844

Since the "inside" of log has to be positive,
x = (-1 + sqrt(161))/2 ~~ 5.844x=1+16125.844 is the answer.