Answer :
To solve the equation [tex]1 + 2\log_{x+2}5 = \log_5(x+2)[/tex], we need to manipulate the equation so we can solve for [tex]x[/tex]. Here's a step-by-step approach:
Start with the given equation:
[tex]1 + 2\log_{x+2}5 = \log_5(x+2)[/tex]Express the '1' as a logarithm with the base of [tex]\log_5(x+2)[/tex]:
[tex]1 = \log_5 5[/tex]This is because the base 5 raised to the power of 1 is 5. So, the equation becomes:
[tex]\log_5 5 + 2\log_{x+2}5 = \log_5(x+2)[/tex]
Use the properties of logarithms:
The property [tex]\log_b a + \log_b c = \log_b (a \cdot c)[/tex] allows us to combine logarithms:[tex]\log_5 5 + \log_{x+2}(5^2) = \log_5(x+2)[/tex]
The 5 inside the second logarithm does not fit well due to different bases, so instead:
[tex]\log_5 5 + \log_{x+2}25 = \log_5(x+2)[/tex]
Simplification here requires understanding a product or base conversion, often assumptions are used in such cases.
Consider properties and switch bases:
Let’s look into bringing terms to the same base:[tex]\log_5(x+2) = \log_5 25[/tex]
Because of symmetry, inspection can show for a solution and easing realizations leads to solving [tex]x+2 = 25[/tex].
Solve for [tex]x[/tex]:
[tex]x+2 = 25 \\
x = 23[/tex]
So, the solution is [tex]x = 23[/tex].