Exponential and Logarithmic Equations Solve the equation. Find the exact

CMIIh

CMIIh

Answered question

2021-09-10

Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible; otherwise, use a calculator to approximate to two decimals.
log8(x+5)log8(x2)=1

Answer & Explanation

pattererX

pattererX

Skilled2021-09-11Added 95 answers

Step 1 
Approach: 
For solving Exponential Equations: 
1: the equation's one side where the exponential expression is isolated
2: Take the logarithm of each side, and then "bring down exponent" using the Laws of Logarithms.
3: Solve for the variable. 
Step 2 
Given, 
log8(x+5)log8(x2)=1 
According to the second logarithm law
loga(AB)=logaAlogaB 
log8(x+5)(x2)=1 
According to the logarithm function's definition
logax=y ay=x 
81=(x+5)(x2) 
8=(x+5)(x2) 
Multiply by (x2) on both sides, 
8(x2)=(x+5)(x2)(x2) 
8(x2)=(x+5) 
As a result of multiplication's distributive feature
8(x)8(2)=(x+5) 
8x16=x+5 
Add (x) on both sides, 
8x16x=x+5x 
7x16=5 
Add 16 on both sides, 
7x16+16=5+16 
7x=21 
Divide by 7 on both sides, 
7x7=217 
x=3 Thus, the exact solution is x=3.

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