Solve the logarithmic equation \ln(x-4)+\ln(x+1)=\ln(x - 8). Be sure to r

skeexerxo175o

skeexerxo175o

Answered question

2021-11-06

Solve the logarithmic equation ln(x4)+ln(x+1)=ln(x8). Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation,correct to two decimal places, for the solution.

Answer & Explanation

Michele Grimsley

Michele Grimsley

Beginner2021-11-07Added 19 answers

Step 1
Given:
ln(x4)+ln(x+1)=ln(x8)
Step 2
It is given that,
ln(x4)+ln(x+1)=ln(x8)
On rewriting the natural logarithm using the base e.
loge(x4)+loge(x+1)=loge(x8)
On expressing the given equation in the form logbM=c
Applying the product rule logb(MN)=logbM+logbN on the left side.
loge[(x4)(x+1)]=loge(x8)
Step 3
It is known that if logbM=logbN then M=N
On using the property to rewriting the equation without logarithms.
(x-4)(x+1)=(x-8)
Simplifying the left side,
x23x4=x8
On rewriting the equation in the standard form
x23x4x+8=x8x+8
x24x+4=0
Step 4
On factorizing x24x+4
(x-2)(x-2)=0
On applying the zero-product principle
x-2=0
x=2
Step 5
On checking the proposed solutions in the original equation,
ln(x4)+ln(x+1)=ln(x8)
ln(24)+ln(2+1)=ln(28)
ln(2)+ln3=ln(6)
The given statement is false
Since the negative numbers do not have logarithms the solution 2 does not check and the given equation has no solution
therefore the solution set is the empty set ϕ

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