004 Sample Final A, Problem 15

Solve. $\log(x+8)+\log(x-1)=1$

Foundations
1) How can we combine the two logs?
2) How do we remove logs from an equation?
Answer:
1) One of the rules of logarithms states that $\log(x)+\log(y)=\log(xy)$
2) The definition of the logarithm tells us that if $\log(x)=y$, then $10^{y}=x$.

Solution:

Step 1:
Using a rule of logarithms, the equation becomes $\log((x+8)(x-1))=1$.

Step 2:
By the definition of the logarithm, $\log((x+8)(x-1))=1$
means $10=(x+8)(x-1)$

Step 3:
Now, we can solve for $x$. We have $0=(x+8)(x-1)-10=x^{2}+7x-18=(x+9)(x-2)$.
So, there are two possible answers, which are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-9} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} .

Step 4:

We have to make sure the answers make sense in the context of the problem. Since the domain of the log function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, \infty)} , -9 is removed as a potential answer. The answer is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} .

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2}

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