Question

question 3
given the equation $4 \cdot 12^{x - 7} = 200$, we want to write $x$ in the form
$x = \frac{\log a + b\log c}{\log c}$

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation \(4\cdot12^{x - 7}=200\) by 4:
\(12^{x - 7}=\frac{200}{4}=50\)

Step2: Take logarithms on both sides

Take the logarithm (we can use common logarithm or natural logarithm, here we use common logarithm) of both sides. Let's take \(\log\) (base 10) of both sides:
\(\log(12^{x - 7})=\log(50)\)

Step3: Use the logarithm power rule

Using the power rule of logarithms \(\log(a^{b}) = b\log(a)\), we can rewrite the left - hand side:
\((x - 7)\log(12)=\log(50)\)

Step4: Solve for \(x\)

First, divide both sides by \(\log(12)\):
\(x - 7=\frac{\log(50)}{\log(12)}\)
Then, add 7 to both sides. We know that \(7=\frac{7\log(12)}{\log(12)}\) (since \(\frac{\log(12)}{\log(12)} = 1\)), so:
\(x=\frac{\log(50)+7\log(12)}{\log(12)}\)
Here, \(a = 50\), \(b = 7\), \(c = 12\)

Answer:

\(x=\frac{\log 50 + 7\log 12}{\log 12}\) (where \(a = 50\), \(b=7\), \(c = 12\))