Question

1)
`e^(5x-9)` = 900

2)
`2^(x-4) =5^(x)`



solve with steps exact answer and approximate answer please!!

Answer 1

The exact and approximate solutions to exponential equations are, respectively: Case 1: Exact:
`x = (9+\ln 900)/(5)`, Approximate: x ≈ 3.160, Case 2: Exact:
`x = \log_{(5)/(2)} (1)/(16)`, Approximate: x ≈ - 3.026.

How to solve exponential equations by algebra properties and definition of logarithms

In this problem we need to find the exact and approximate solutions to exponential equations, which can be found by definition of logarithms and algebra properties:

Case 1:

e⁵ˣ⁻⁹ = 900

5 · x - 9 = ㏑ 900

5 · x = 9 + ㏑ 900

`x = (9+\ln 900)/(5)` (Exact)

x ≈ 3.160 (Approximate)

Case 2:

2ˣ⁻⁴ = 5ˣ

`(2^x)/(16) =5^x`

`\left((5)/(2) \right)^x=(1)/(16)`

`x \cdot \log_{(5)/(2)} (5)/(2) = \log_{(5)/(2)} (1)/(16)`

`x = \log_{(5)/(2)} (1)/(16)` (Exact)

x ≈ - 3.026 (Approximate)