Question

Solve:
5^x-1 + 5^x-2 +5^ x-3 =153​

Answer 1

`x\approx3.992`. exact form is
`x=\log_5(153)-\log_5(31)+3`

Explanation:

we are given the equation
`5^(x-1)+5^(x-2)+5^(x-3)=153`.

first, we will use the exponent property:
`a^(m-n)=(a^m)/(a^n)`.

so, we have
`(5^x)/(5)+(5^x)/(5^2)+(5^x)/(5^3)=153`. we will simplify this as:
`(5^x)/(5)+(5^x)/(25)+(5^x)/(125)=153`. next, we will convert the fractions into ones with the same denominator.

`\to (5^x*25)/(125)+(5^x*5)/(125)+(5^x)/(125)` then we add them together.

`\to((25+5+1)5^x)/(125)=(31(5^x))/(125)=31(5^(x-3))=153`

next, we divide both sides by 31.

`5^(x-3)=(153)/(31)`

next, we find the logarithm of each side, with base 5. note that
`\log_a(a^n)=n`.

so,
`x-3=\log_5((153)/(31))`

and finally, we have
`x=\log_5((153)/(31))+3\approx3.992`.

we can simplify the exact form using
`\log((a)/(b))=\log(a)-\log(b)`

so, we can have
`x=\log_5(153)-\log_5(31)+3`

Answer 2

Let's solve the equation 5^(x-1) + 5^(x-2) + 5^(x-3) = 153.

To solve this equation, we can notice that all the terms on the left side have a common base of 5. We can rewrite the equation as follows:

(5^1) * 5^(x-1) + (5^2) * 5^(x-2) + (5^3) * 5^(x-3) = 153

Now, we can simplify the equation:

5^1 * 5^(x-1) + 5^2 * 5^(x-2) + 5^3 * 5^(x-3) = 153

5^(x-1+1) + 5^(x-2+2) + 5^(x-3+3) = 153

5^x + 5^x + 5^x = 153

3 * 5^x = 153

Now, we can divide both sides of the equation by 3:

5^x = 51

To solve for x, we can take the logarithm of both sides with base 5:

log base 5 (5^x) = log base 5 (51)

x = log base 5 (51)

Using a calculator, we find that x is approximately 2.357.

So, the solution to the equation 5^(x-1) + 5^(x-2) + 5^(x-3) = 153 is x ≈ 2.357.