Exponential Problem Solving

Exponential Problem Solving-38
Just remember when exponential functions are involved, functions are increasing or decreasing very quickly (multiplied by a fixed number).That’s why it’s really good to start saving your money early in life and let it grow with time.When \(b) many times by itself, it gets smaller, since the denominator gets larger.

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We are going to treat these problems like any other exponential equation with different bases--by converting the bases to be the same.

Whether we like it or not, we need to revisit exponents and then start talking about logs, which will help us solve exponential and logarithmic equations.

It's an equation that has exponents that are $$ \red$$.

$$ \red 4^3 = \red 2^x $$ $$ \red 9^x = \red $$ $$ \left( \red \right)^ = \red 4^3 $$ $$ \red 4^ 1 = \red $$ In each of these equations, the base is different.

Remember that exponential functions are named that because of the “\(x\)” in their exponents! “\(b\)” is called the base of the exponential function, since it’s the number that is multiplied by itself “\(x\)” times (and it’s not an exponential function when \(b=1\)).

\(b\) is also called the “growth” or “decay” factor.

These are vertical transformations or translations. When transformations are made on the inside of the \(f(x)\) part, you move the function back and forth (but do the opposite math – basically since if you were to isolate the \(x\), you’d move everything to the other side).

These are horizontal transformations or translations. When there is a negative sign outside the parentheses, the function is reflected (flipped) across the \(x\)-axis; when there is a negative sign inside the parentheses, the function is reflected across the \(y\)-axis.

In this tutorial, learn how to turn a word problem into an exponential decay function.

$$ 2^ = 4 \ 8^ = 16 \ \ 16^ = 256 \ \left( \frac \right)^ = 512 $$ As you might've noticed, an exponential equation is just a special type of equation.


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