Solve the following equations: b) 5^x - 6 imes 5^x+1 + 125 = 0. [Also, 3^x imes 5^x imes 7^x = 11,0252x+ is…

Question

Solve the following equations: b) 5^x - 6 imes 5^x+1 + 125 = 0. [Also, 3^x imes 5^x imes 7^x = 11,0252x+ is incomplete]

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the equations:

8. Solve each of the following equations

b) $5^x - 6 \times 5^{x+1} + 125 = 0$

Use the exponent rule $a^{m+n} = a^m \times a^n$ to rewrite $5^{x+1}$ .
$5^{x+1} = 5^x \times 5^1 = 5 \times 5^x$
Substitute this back into the equation.
$5^x - 6 \times (5 \times 5^x) + 125 = 0$ $5^x - 30 \times 5^x + 125 = 0$
Combine the terms involving $5^x$ .
$(1 - 30) \times 5^x + 125 = 0$ $-29 \times 5^x + 125 = 0$
Isolate the term with $5^x$ .
$-29 \times 5^x = -125$ $5^x = \frac{-125}{-29}$ $5^x = \frac{125}{29}$
Solve for $x$ using logarithms.
$x = \log_5\left(\frac{125}{29}\right)$
Apply the logarithm property $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$ .
$x = \log_5(125) - \log_5(29)$
Simplify $\log_5(125)$ since $125 = 5^3$ .
$x = 3 - \log_5(29)$ \boxed{\text{x = 3 - \log_5(29)}}
The second equation, $3^x \times 5^x \times 7^x = 11,025^{2x+}$ , is incomplete as the exponent on the right side is cut off. Please provide the full equation for me to solve it.

Solve the following equations: b) 5^x - 6 imes 5^x+1 + 125 = 0. [Also, 3^x imes 5^x imes 7^x = 11,0252x+ is incomplete]