# Video: Solving Logarithmic Equations over the Set of Real Numbers

Solve logβ [251 + logβ (π₯ + 7)] = 8, where π₯ β β.

02:39

### Video Transcript

Solve the equation log base two of 251 plus log base three of π₯ plus seven equals eight, where π₯ is an element of the real numbers.

Here, weβve been given a logarithmic equation. This equation involves logs with two different bases. Weβve got log base two over here and log base three over here. Now, we recall that a logarithm is the power to which a number must be raised in order to get some other number.

Letβs take a general logarithm. Letβs say log base π of π is equal to π. Now, taking log base π is essentially the inverse of raising to the power of π. So we raise both sides as a power of π. And we get π to the power of log base π of π equals π to the power of π. And since taking the log base π is the inverse of raising it as a power of π, the left-hand side simply becomes π. And so saying log base π of π is equal to π is equivalent to saying that π is equal to π to the power of π.

And so we can consider solving our equation in two ways. We could use this general definition, or we could raise both sides as a power of two. When we do so, on the left-hand side, this is the equivalent of the letter π in our general form. Weβre left with 251 plus log base three of π₯ plus seven. And this is equal to two to the eighth power in our general form. Thatβs π to the power of π. Two to the eighth power is 256. So our equation is 251 plus log base three of π₯ plus seven equals 256. Letβs subtract 251 from both sides. And when we do, we see that log base three of π₯ plus seven is equal to five.

And now weβre going to perform a similar step to our first step here. We could either use the general definition or equivalently raise both sides. But this time, we do it as a power of three. When we do so, on the left-hand side, weβre left with simply π₯ plus seven. And of course, once again, this is equivalent to π in the general definition. On the right-hand side, we get three to the fifth power. Three to the fifth power is 243. So our equation becomes π₯ plus seven equals 243.

Remember, weβre solving for π₯. So, finally, we just subtract seven from both sides. 243 minus seven is 236. And so weβve solved the equation. We get π₯ equals 236. And of course, we could check the solution depending on the functionality of our calculator by substituting π₯ is equal to 236 into our original expression and checking we do indeed get eight.