Question

Solve. 5²ˣ = 21 a. 0.786 b. 0.879 c. 0.946 d. 0.649 e. 0.196

Answer 1

Answer: 0.946

Step-by-step explanation

The variable is in the trees, so we must log it down.

`5^{2\text{x}} = 21\\\\\log\left(5^{2\text{x}}\right) = \log(21)\\\\2\text{x}\log\left(5\right) = \log(21)\\\\\text{x} = (\log(21))/(2\log\left(5\right))\\\\\text{x} \approx (1.322219)/(1.397940)\\\\\text{x} \approx 0.945834\\\\\text{x} \approx 0.946\\\\`

Each log mentioned above is base 10.

I used the log rule
`\log(A^B) = B\log(A)` on the third step. This log rule is very important. This rule allows us to isolate the exponent.

Answer 2

The value of x that satisfies the equation 5^2x = 21 is approximately 0.879 (Option b).

Explanation:

To find the value of x, we'll use logarithms to solve for x in the equation 5^2x = 21. Taking the logarithm base 5 of both sides helps isolate x.

log_5(5^2x) = log_5(21)

2x = log_5(21)

x = log_5(21) / 2

Now, calculate log_5(21) and divide it by 2 to find the value of x.

log_5(21) ≈ 2.302

x ≈ 2.302 / 2

x ≈ 1.151 / 2

x ≈ 0.879

Therefore, x ≈ 0.879, which corresponds to option b. This value satisfies the equation 5^2x = 21. When 5 is raised to the power of approximately 0.879, it equals 21. This method uses logarithms to solve exponential equations and find the unknown variable x.