Solve equation, using logarithms?

[dolphin_girl93]

How would you solve these equations using common logarithms?

3^(x+4) = 2^(1-3x)

5^x + 125(5^-x)=30?



I thought for the first one I was supposed to equate bases but I don't know how you'd do that. So is that and I'm just not seeing it or something else?

For the second one again I thought you might equate bases but I think the 125(5^-x) is throwing me off?

 

[Mrspi]

How would you solve these equations using common logarithms?

3^(x+4) = 2^(1-3x)

5^x + 125(5^-x)=30?



I thought for the first one I was supposed to equate bases but I don't know how you'd do that. So is that and I'm just not seeing it or something else?

For the second one again I thought you might equate bases but I think the 125(5^-x) is throwing me off?

for the first problem, take the log of both sides of the equation:

log [3^(x + 4)] = log [2^(1 - 3x)]

Now use the rules of logs.

log (a^b) = b*log a

So, log[3^(x + 4)] = (x + 4)*log 3

And log [2^(1 - 3x)] = (1 - 3x) * log 2

Now you have this:

(x + 4)*log 3 = (1 - 3x) * log 2

Use the distributive property:

x*log 3 + 4*log 3 = log 2 - 3x*log 2

From here on out, it is just "normal" equation solving tactics. Get all of the terms containing "x" on one side of the equation, and all terms that do NOT contain x on the other side.

x*log 3 + 3x*log 2 = log 2 - 4 log 3

Factor "x" out of both terms on the left side:

x*(log 3 + 3 log 2) = log 2 - 4 log 3

Divide both sides by (log 3 + 3 log 2):

x = [log 2 - 4 log 3] / [log 3 + 3 log 2]

Now, it's calculator time.

 

[Deleted member 4993]

How would you solve these equations using common logarithms?

3^(x+4) = 2^(1-3x)

ln[3^(x+4)] = ln[2^(1-3x)]

(x+4) * ln[3] = (1-3x) * ln(2}

Now continue.....

5^x + 125(5^-x)=30?

5^x + 5^3 * 5^(-x) = 5 * 6



I thought for the first one I was supposed to equate bases but I don't know how you'd do that. So is that and I'm just not seeing it or something else?

For the second one again I thought you might equate bases but I think the 125(5^-x) is throwing me off?