Solve the following exponential equations: 1. 2^x = 2 2. 52x+1 = 125 3. x3 = 64 4. 2(3^x) = 18 5. 3 imes…

Question

Solve the following exponential equations: 1. 2^x = 2 2. 52x+1 = 125 3. x3 = 64 4. 2(3^x) = 18 5. 3 imes 52x+4 = 75

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

1. $2^x = 2$

Rewrite the right side with the same base as the left side.
$2^x = 2^1$
Since the bases are the same, equate the exponents.
$x = 1$
The solution is $\boxed{\mathbf{x=1}}$ .

2. $5^{2x+1} = 125$
Express 125 as a power of 5.
$5^{2x+1} = 5^3$
Since the bases are the same, equate the exponents.
$2x+1 = 3$
Solve the linear equation for $x$ .
$2x = 3 - 1$ $2x = 2$ $x = \frac{2}{2}$ $x = 1$
The solution is $\boxed{\mathbf{x=1}}$ .

3. $x^3 = 64$
Express 64 as a cube of an integer.
$x^3 = 4^3$
Take the cube root of both sides.
$x = 4$
The solution is $\boxed{\mathbf{x=4}}$ .

4. $2(3^x) = 18$
Divide both sides by 2.
$\frac{2(3^x)}{2} = \frac{18}{2}$ $3^x = 9$
Express 9 as a power of 3.
$3^x = 3^2$
Since the bases are the same, equate the exponents.
$x = 2$
The solution is $\boxed{\mathbf{x=2}}$ .

5. $3 \times 5^{2x+4} = 75$
Divide both sides by 3.
$\frac{3 \times 5^{2x+4}}{3} = \frac{75}{3}$ $5^{2x+4} = 25$
Express 25 as a power of 5.
$5^{2x+4} = 5^2$
Since the bases are the same, equate the exponents.
$2x+4 = 2$
Solve the linear equation for $x$ .
$2x = 2 - 4$ $2x = -2$ $x = \frac{-2}{2}$ $x = -1$
The solution is $\boxed{\mathbf{x=-1}}$ .

Solve the following exponential equations: 1. 2^x = 2 2. 52x+1 = 125 3. x3 = 64 4. 2(3^x) = 18 5. 3 imes 52x+4 = 75