What's the logarithmic equation 2=log(3x+4)

Question

zepdrix · Answer

Gimme one sec, I got somethin for ya

anonymous · Answer

All logs without a base written in are base 10, and logs are like the inverses of exponents

zepdrix · Answer

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zepdrix · Answer

Are you able to solve the problem once it's in exponential form?

anonymous · Answer

Thank you so much for that, @zepdrix! I usually have trouble when it comes to putting it in exponential form and then solving. However, I'll see what I can do.

        10^2=3x+4
              so
          100=3x+4
         subtract 4 from 100 and 4
           96=3x
           divide by three on both sides.
              So, my answer would be 32?

anonymous · Answer

Could either of you maybe explain to me how we would convert 5=log3 (x^2+18) or e^x e^(x+1)=1 into an exponential equation and then solve?

zepdrix · Answer

Try to draw the arrows like I was doing before! :O

So the base 3 points to the 5, then the 5 points to the x^2+18.
It's like a spiral that goes around.

So the exponential form is:

3^5 = x^2+18

zepdrix · Answer

So the other equation:

e^x * e^(x+1) = 1

This one is uhhhhh a little tricky.

So we'll apply a rule of exponents:
When we multiply terms of similar bases, we ADD the exponents.

So our left side becomes:
e^(x+(x+1)) = 1

We can then rewrite 1 in terms of the exponential of base e.

e^0 = 1 let's write our right side like this.

e^(x+x+1) = e^0

zepdrix · Answer

The bases are the same, which means the exponents have to also be equal.
Set them equal to one another.

x+x+1 = 0

Then it's pretty straight forward solving for x from there.

anonymous · Answer

Yes! Thank you so much for all your help!! Just to make sure, the final answer to that problem is x= -1/2, right? @zepdrix

mathmale · Answer

I'd like to present a different point of view:

As samarasamara has pointed out, All logs without a base written in are base 10, and logs are like the inverses of exponents

These two facts are instrumental in solving this present problem.

Given 2 = log (3x+4) and the equation 10 = 10,
Let 2 be the exponent of the left 10 and let log(3x+4) be the exponent of the right 10.
                            2  
This reduces to 10    = 3x+4, or 100=3x+4, or 3x=96, or x=??  

that's the solution.

zepdrix · Answer

-1/2? Ya looks good!

mathmale · Answer

zepdrix:  perhaps we'd better each check our respective "answers" in  the original equation.

mathmale · Answer

log (3[32]+4) = log 100 = log 10^ 2 = 2.

log (3[-1/2]+4) = log (1/2) = log 1 - log 2 = 0 - log 2, or approx. -0.693.