Find the exact solution if possible: \log_{3}(x-8)+\log_{3}x=2

melodykap

melodykap

Answered question

2021-08-03

Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.
log3(x8)+log3x=2

Answer & Explanation

krolaniaN

krolaniaN

Skilled2021-08-04Added 86 answers

Step 1
For solving Exponential Equations:
1) Isolate the exponential expression on one side of the equation.
2) Take the logarithm of each side, then use the Laws of Logarithms to "bring down exponent".
3) Solve for the variable.
Step 2
Given,
log3(x8)+log3x=2
By the first law of logarithms function,
log3(x8)x=2
By the definition of logarithm function,
logax=y ay=x
32=x(x8)
9=x(x8)
By the distributive property of multiplication,
x xx8=9
x28x=9
Add 9 on both sides,
x28x9=99
x28x9=0
Factor x28x9 as (x9)(x+1).
(x9)(x+1)=0
By the zero-factor property,
x9=0 and x+1=0
Taking,
x9=0
Add 9 on both sides,
x9+9=0+9
x=9
Now taking,
x+1=0
Add 1 on both sides,
x+11=01
x=1
Here, we neglect x=1 because logarithm function cannot be negative.
Thus, the exact solution is x=9.

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