# Solve exponential equations with different bases in a relationship

### Logs And Exponentials

Logarithms can be used to solve equations such as 2x = 3, for x. In senior We now seek to give meaning to other types of exponents. The basic principle we use .. The relationship connecting logarithms and powers is: x = loga y means y . An exponential equation is one in which a variable occurs in the exponent. the same base can be solved using this property: Use this inverse relationship. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay. This relationship is true for any function and its inverse. which all follow from identities involving exponents and the definition of the logarithm.

### The exponential function | Maths | Education in Chemistry

And this is pretty straightforward to solve. Subtract five from both sides. And we get 9x is equal to negative five.

Divide both sides by nine, and we are left with x is equal to negative five.

Let's do another one of these, and let's make it a little bit more, a little bit more interesting. Let's say we have the exponential equation two to the 3x plus five power is equal to 64 to the x minus seventh power. Once again, pause the video, and see if you can tell me what x is going to be, or what x needs to be to satisfy this exponential equation.

All right, so you might, at first, say, oh, wait a minute, maybe 3x plus five needs to be equal to x minus seven, but that wouldn't work, because these are two different bases.

You have two to the 3x plus five power, and then you have 64 to the x minus seven. So, the key here is to express both of these with the same base, and lucky for us, 64 is a power of two, two to the, let's see, two to the third is eight, so it's going to be two to the third times two to the third. Eight times eight is 64, so it's two to the sixth is equal to 64, and you can verify that.

Take six twos and multiply them together, you're going to get This was just a little bit easier for me. When does five times two to the t power equal 1, Whenever we're doing anything algebraically it's always a little bit useful to see if we can isolate the variable that we're trying to solve for, we're trying to find what t value will make this equal that right over there.

A good first step would maybe try to get this five out of the left hand side, so let's divide the left by five. If we want to keep this being in equality we have to do the same thing to both sides.

We get two to the t power is equal to 1, over five.

## Exponential Equations in Science I: Growth and Decay

How do we solve for t here? What function is essentially the inverse of the exponential function? Well it would be the logarithm.

If we say that a to the b power is equal to c then that means that log base a of c is equal to b. Log base a of c says what power do I need to raise a to, to get to c? Well I need to raise a to the b power, to get to c.

• Exponentials & logarithms
• The exponential function
• Solving Exponential Equations

These two are actually equivalent statements. Let's take log base two of both sides of this equation. On the left hand side you have log base two of two to the t power. A few years later, the Coast Guard abandoned the island, leaving the reindeer behind. Inresearcher Dave Klein, a biologist working for U. Fish and Wildlife Service, visited the island and counted 1, healthy reindeer, whose population had exploded due to the lack of predators and the abundance of lichenstheir primary food source.

Klein returned to the island inwhere he was astounded to find that the population had swelled to over 6, which represents an amazing 47 reindeer per square mile.

Klein, Professor emeritus, University of Alaska Fairbanks.

### The exponential function | Maths | Education in Chemistry

Three years later inKlein and others returned to the island to find that the reindeer population had plummeted from 6, healthy reindeer to 42 sickly reindeer. Of those 42, 41 were females and 1 was a male that had abnormal antlers, a sign that it was probably unable to reproduce. Overgrazing had wiped out the lichen supply, a significant winter food source for the reindeer.

Matthew Island Klein, By the s, no reindeer remained on the island. Notice that the population of reindeer on the island changed relatively slowly from year to year in the beginning, but as time went on, it increased by larger and larger amounts.