Exponential Equation: Solve 4x+7=18

[DTA]

Homework Statement

Solve for x:

4^x+7=18

Homework Equations

It becomes 4^x=11

The Attempt at a Solution

I have no idea where to go from here. We never had a problem with such an immovable number such as 11 to work with. Calculators are not allowed to be used either.

 

[Dickfore]

Use the definition of a logarithm.

 

[DTA]

Okay, I think I'm on the right track but I'm still not sure where to go with this:

4^log₄11= ?

 

[Dickfore]

Okay, I think I'm on the right track but I'm still not sure where to go with this:

4^log₄11= ?

For logarithms it is always true that:

$$a^{x} = b \Leftrightarrow x = \log_{a}{b}, \; a, b > 0, \; a \neq 1$$

From here we have the fundmanetal property of logarithms:

$$a^{\log_{a} b} = b$$

 

[LCKurtz]

It becomes 4^x=11

Take the log of both sides to any base and solve for x. It doesn't have to be base 4 and if you were after a decimal approximation you would likely want to use base e or base 10.

 

[DTA]

Okay, so I attempted a change of base and got this:

ln11/ln4 ~ 1.729

Is this how it eventually would end up?


Thanks for all your help! Logarithms are just driving me nuts for some reason

 

[Dickfore]

yes.