Exponential Equation: Solve 4x+7=18

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To solve the equation 4x + 7 = 18, first isolate the variable by rewriting it as 4x = 11. The next step involves using logarithms, specifically the property that a^x = b implies x = log_a(b). Taking the logarithm of both sides allows for solving for x, and while any base can be used, base e or base 10 is preferable for decimal approximations. The final calculation yields approximately ln(11)/ln(4) ≈ 1.729, which is the solution for x. Logarithmic concepts can be challenging, but they are essential for solving such equations.
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Homework Statement



Solve for x:

4x+7=18

Homework Equations



It becomes 4x=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.
 
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Use the definition of a logarithm.
 
Okay, I think I'm on the right track but I'm still not sure where to go with this:

4log411= ?
 
DTA said:
Okay, I think I'm on the right track but I'm still not sure where to go with this:

4log411= ?

For logarithms it is always true that:

<br /> a^{x} = b \Leftrightarrow x = \log_{a}{b}, \; a, b &gt; 0, \; a \neq 1<br />

From here we have the fundmanetal property of logarithms:

<br /> a^{\log_{a} b} = b<br />
 
DTA said:
It becomes 4x=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.
 
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
 
yes.
 
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