Question

Explain how to solve 5^(x-2)= 8 using the change of base formula

Answer 1

its actually 3.292 because we round to the nearest thousandth and thats not even the equation you use above

Explanation:

For this equation we use the formula log a^m=m (log a) so the equation will be written as log 5 (5^x-2) = log 5 (8). You use the base, which is 5, and use log to base 5 on both sides of the equation. Then you take the exponent " x-2" and write( x-2) log 5 (5) = log 5(8). Since log a =1, you multiply that 1 by x-2, which keeps it x-2. Making the equation x-2 = log 5 (8). Next, we use the change of the base properties with the formula log b^y= log y/ log b. The equation will be written as x-2 = log 8/ log 5, since 5 is the base it stays in the bottom or basement. We then add +2 to both sides of x-2 and log 8/ log 5. To solve this equation, you can find out what log 8 and log 5 are and divide those and add +2 to solve. So log 8 = 0.903 and log 5 = 0.698970 and divide those to get 1.29190 +2 and you get the answer rounded as 3.292.

Answer 2

x = 3.3

Explanation:

A equation is given to us and we need to solve out for x. The given equation is ,

`\sf\longrightarrow 5^(x -2)= 8`

Take log on both sides with base as " 10" . We have ,

`\sf\longrightarrow log_(10) 5^(x-2)= log_(10)\ 8`

Simplify using the property of log ,
`\sf log a^m = m log a` , we have ,

`\sf\longrightarrow ( x -2) log_(10) 5 = log_(10) 8`

Simplify ,

`\sf\longrightarrow ( x -2 ) log_(10)5 = log_(10) 2^3`

Again simplify using the property of log ,

`\sf\longrightarrow (x-2) log 5 = 3 log 2`

We know that log 5 = 0.69 and log 2 = 0.301 , on substituting this , we have ,

`\sf\longrightarrow ( x - 2 ) = ( 3* 0.301)/(0.69)`

Simplify the RHS ,

`\sf\longrightarrow x - 2 = 1.30`

Add 2 both sides ,

`\sf\longrightarrow \boxed{\blue{\sf x = 3.30}}`

Hence the Value of x is 3.30 .