anonymous
  • anonymous
Please help. I don't want the answer just the steps to solve it myself thanks so much. Solve for x and express your answer as a logarithm. 3(5^x) = 48 Simplify your answer as much as possible.
Mathematics
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anonymous
  • anonymous
Please help. I don't want the answer just the steps to solve it myself thanks so much. Solve for x and express your answer as a logarithm. 3(5^x) = 48 Simplify your answer as much as possible.
Mathematics
schrodinger
  • schrodinger
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marco26
  • marco26
just use \[a ^{x}=y\] ->\[\log _{a}y=x\]
Owlcoffee
  • Owlcoffee
\[(3)(5^x)= 48\] When we want to solve an exponential equation, called like this because the variable right now is in the exponent, we have to use the reciprocate operation in order to bring the x down as a normal variable and not an exponent. What is reciprocate to exponents?, well Logarithms. In essence, we can use any logarithm we want, but we always take the logarithm with the convenient base in order to make things simple, so, we will can begin by getting rid of the 3 by dividing both sides by 3: \[5^x=\frac{ 48 }{ 3 }\] \[5^x=16\] So, now, what we will do is take the logarithm with base 5, which means: \(\log_{5} \) on both sides of the equation, further on, use the property of logarithms \(\log_{a} b^n=n \log_{a} b\) to bring the variable "x", down from the exponent: \[5^x=16\] \[\log_{5} 5^x=\log_{5} 16\] \(x \log_{5} 5=\log_{5} 16\) ...and \(\log_{5} 5=1\) so therefore: \[x=\log_{5} 16\]
anonymous
  • anonymous
Thank you so much for taking the time out of your day to help me it it means a lot.

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anonymous
  • anonymous
Owlcoffee
  • Owlcoffee
No problem
Jhannybean
  • Jhannybean
\[3(5^x) = 48\]Dividing both sides by 5, we would get \[5^x = \frac{48}{3} \qquad \implies 5^x = 16\] and now we can take the natural log of both sides to find x. \[\log_5 (5^x) = \log_5 (16) \]\[x = \log_5(16)\] Using the change of base formula we can simplify \(\log_5(16) \) into a fractional exponent. Remember that \(\log_a(x) = \dfrac{\log_b(x)}{\log_b(a)}\) Applying this you would get \[\color{red}{\boxed{x=\log_5(16) = \frac{\log(2^4)}{\log(5)}=\frac{4\log(2)}{\log(5)}}}\]
Jhannybean
  • Jhannybean
dividing both sides by 3*

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