Sure! Let's solve the equation [tex]\(5^{x+2} = 14^x\)[/tex] step-by-step.
### Step 1: Understand the equation
We have [tex]\(5^{x+2}\)[/tex] on the left side and [tex]\(14^x\)[/tex] on the right side. To solve for [tex]\(x\)[/tex], we'll need to use logarithms.
### Step 2: Apply logarithms
Taking the logarithm of both sides (we can use any logarithm base, but we'll use the natural logarithm, [tex]\(\ln\)[/tex], for simplicity):
[tex]\[
\ln(5^{x+2}) = \ln(14^x)
\][/tex]
### Step 3: Simplify using logarithm properties
Use the property [tex]\(\ln(a^b) = b\ln(a)\)[/tex]:
[tex]\[
(x+2)\ln(5) = x\ln(14)
\][/tex]
### Step 4: Distribute and rearrange
Distribute the [tex]\(\ln(5)\)[/tex]:
[tex]\[
x\ln(5) + 2\ln(5) = x\ln(14)
\][/tex]
Get all terms involving [tex]\(x\)[/tex] on one side of the equation:
[tex]\[
x\ln(5) - x\ln(14) = -2\ln(5)
\][/tex]
Factor out [tex]\(x\)[/tex] on the left side:
[tex]\[
x(\ln(5) - \ln(14)) = -2\ln(5)
\][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Divide both sides by [tex]\((\ln(5) - \ln(14))\)[/tex]:
[tex]\[
x = \frac{-2\ln(5)}{\ln(5) - \ln(14)}
\][/tex]
### Step 6: Simplify the expression
Knowing the property [tex]\(\ln(a/b) = \ln(a) - \ln(b)\)[/tex]:
[tex]\[
x = \frac{-2\ln(5)}{\ln \left( \frac{5}{14} \right)}
\][/tex]
### Step 7: Final expression
This simplifies to:
[tex]\[
x = \frac{-2\ln(5)}{\ln \left( 5/14 \right)}
\][/tex]
Applying the properties of exponentials and logarithms, the solution simplifies down further to represent the same form you got, which can be written explicitly with the logarithm base and power properties:
[tex]\[
x = -\log{\left(5^{2 / \log{\left(\frac{5}{14}\right)}}\right)}
\][/tex]
Thus, the solution to the equation [tex]\(5^{x+2} = 14^x\)[/tex] is:
[tex]\[
x = -\log{\left(5^{2 / \log{\left(\frac{5}{14}\right)}}\right)}
\][/tex]
This matches the result we obtained previously and hence is the solution for this equation.
