Question

Mike is working on solving the exponential equation 74^x = 37; however, he is not quite sure where to start. Using complete sentences, describe to Mike how to solve this equation.

Answer 1

`x = \log_(74)(37)`

Explanation:

we want to solve the following exponential equation:

`\iff {74}^(x) = 37`

Taking common logarithm of both sides yields:

`\iff \log{74}^(x) = \log37`

recall that,

• 
  `log( {a}^(x) ) = x log(a)`

utilizing it yields:

`\iff x \cdot\log{74} = \log37`

divide both sides by log74:

`\iff \frac{ x \cdot\log{74}}{ \log(74)} = ( \log37)/( \log(74) )`

remember that,

`log_b( {a} ) = (log(a))/(log(b))`

hence,

`x = \boxed{ \log_(74)(37)}`

Answer 2

Method 1

`74^x = 37`

Take natural logs of both sides:

`\implies \ln74^x = \ln37`

Apply the log power rule:
`\ln(a)^b=b \ln (a)`

`\implies x\ln74 = \ln37`

Divide both sides by ln 74:

`\implies (x\ln74)/(\ln74) = (\ln37)/(\ln74)`

`\implies x= (\ln37)/(\ln74)`

`\implies x=0.8389552282...`

Method 1

`74^x = 37`

Take logs of base 74 of both sides:

`\implies \log_(74)74^x = \log_(74)37`

Apply the log power rule:
`\log_am^n=n\log_am`

`\implies x\log_(74)74= \log_(74)37`

Apply log of the same number as base rule:
`\log_aa=1`

`\implies x= \log_(74)37`

`\implies x=0.8389552282...`