Question 883007
Hello!
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You can use a few "logarithm rules" to expand out logarithms of any base:
1. The log of the product of two numbers is the same as adding the individual logs of each number: {{{log (a, (bc)) = log (a, b) + log (a, c)}}}
2. The log of the quotient of two numbers is the same as subtracting the individual logs of each number: {{{log(a,(b/c)) = log(a,b) - log(a,c)}}}
3. The log of a number raised to a certain exponent is the same as the value of that exponent multiplied by the log of that number: {{{log(a,(b^c)) = c*log(a,b)}}}
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Now let us apply those rules to this case!
{{{ln((x^3 * (x - 2)^2)/(sqrt(x^2+5)))}}}
Use rule #2 to get rid of the division: {{{ln(x^3 * (x-2)^2) - ln(sqrt(x^2 + 5))}}}
Use rule #1 to get rid of the multiplication: {{{ln(x^3) + ln((x-2)^2) - ln(sqrt(x^2 + 5))}}}
Since taking the square root is the same thing as raising to the one-half power: {{{ln(x^3) + ln((x - 2)^2) - ln((x^2 + 5)^(1/2))}}}
Use rule #3 to get rid of exponents: {{{3ln(x) + 2ln(x - 2) - (1/2)ln(x^2 + 5)}}}
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Therefore, the expanded expression is {{{3ln(x) + 2ln(x - 2) - (1/2)ln(x^2 + 5)}}}
I hope this helps! =)