Solve the equation log base two of 251 plus log base three of 𝑥 plus seven equals eight, where 𝑥 is an element of the real numbers.
Here, we’ve been given a logarithmic equation. This equation involves logs with two different bases. We’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.
Let’s take a general logarithm. Let’s say log base 𝑏 of 𝑎 is equal to 𝑐. Now, taking log base 𝑏 is essentially the inverse of raising to the power of 𝑏. So we raise both sides as a power of 𝑏. And we get 𝑏 to the power of log base 𝑏 of 𝑎 equals 𝑏 to the power of 𝑐. And since taking the log base 𝑏 is the inverse of raising it as a power of 𝑏, the left-hand side simply becomes 𝑎. And so saying log base 𝑏 of 𝑎 is equal to 𝑐 is equivalent to saying that 𝑎 is equal to 𝑏 to the power of 𝑐.
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 𝑎 in our general form. We’re left with 251 plus log base three of 𝑥 plus seven. And this is equal to two to the eighth power in our general form. That’s 𝑏 to the power of 𝑐. Two to the eighth power is 256. So our equation is 251 plus log base three of 𝑥 plus seven equals 256. Let’s subtract 251 from both sides. And when we do, we see that log base three of 𝑥 plus seven is equal to five.
And now we’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, we’re left with simply 𝑥 plus seven. And of course, once again, this is equivalent to 𝑎 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 𝑥 plus seven equals 243.
Remember, we’re solving for 𝑥. So, finally, we just subtract seven from both sides. 243 minus seven is 236. And so we’ve solved the equation. We get 𝑥 equals 236. And of course, we could check the solution depending on the functionality of our calculator by substituting 𝑥 is equal to 236 into our original expression and checking we do indeed get eight.