Logarithmic

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Backwards

What if we have a function value for an exponential function and we would like to know which domain numbers are associated with it? In other words, we would like to solve

\[ g(t) = A \cdot r^t = g_0 \]

How would we solve for $t$?

If the function value “works well”, then we could probably guess.

Let $T(f) = 4 \cdot 3^f$.

Solve $T(f) = 36$

$4 \cdot 3^f = 36$

$3^f = 9$

$f = 2$

Most function values are not going to be so obvious.

For example, solve $ 3^x = 17$.

We may not be able to quickly think up this number or even an approximation for it. However, we can still talk about it.

We are looking for the number that you raise $3$ to, to get $17$.

That is a specific number. We can see from the graph that there is only one such number and we could visually approximate it around $2.6$.

We can see from the general shape of the graph of an exponential function that any function value is attached to a single unique domain number.

The following are all descriptions that identify unique real numbers.

The number that you raise $5$ to, to get $97$.
The number that you raise $\frac {3}{4}$ to, to get $6$.
The number that you raise $7$ to, to get $\frac {1}{2}$.
The number that you raise $101$ to, to get $34$.
The number that you raise $10$ to, to get $1,000$.

As with all mathematical phrases, we have shorthand notation for these desciptions.

Logarithm Base A of B

Let $a$ and $b$ be positive real numbers. The number you raise $a$ to, to get $b$ is called the logarithm base a of b.

The symbol for the logarithm base a of b is $\log _a(b)$.

$\log _a(b)$ is the number you raise $a$ to, to get $b$.

\[ a^{\log _a(b)} = b \]

The following are all descriptions that identify unique real numbers.

The number that you raise $5$ to, to get $97$ is $\log _5(97)$.
The number that you raise $\frac {3}{4}$ to, to get $6$ is $\log _{\tfrac {3}{4}}(6)$.
The number that you raise $7$ to, to get $\frac {1}{2}$ is $\log _7\left (\frac {1}{2}\right )$.
The number that you raise $101$ to, to get $34$ is $\log _{101}(34)$.
The number that you raise $10$ to, to get $1,000$ is $\log _{10}(1000)$.

Evaluate the following expressions

$3^{\log _3{56}} = \answer {56}$
$13^{\log _{13}{21}} = \answer {21}$
$\pi ^{\log _{\pi }{82}} = \answer {82}$
$4^{\log _4{\sqrt {7}}} = \answer {\sqrt {7}}$
$85^{\log _{85}{2}} = \answer {2}$

Logarithms are exponents. They can be positive or negative.

$9^{-2} = \frac {1}{81}$, therefore $\log _{9}\left (\frac {1}{81}\right ) = -2$

We also know that raising a positve number to any exponent cannot produce a negative number or $0$. Therefore, the number inside the logarithm must be positive

It sounds like we have a new category of functions.

Basic Logarithmic Functions

A Basic Logarithmic Function is a function that can be represented by formulas of the form

\[ L(x) = \log _r(x) \]

where $r > 0$.

The domain is positive real numbers and the range is all real numbers.

Here is the graph of $y = L(x) = \log _2(x)$.

The intercept is $(1,0)$, because $\log _2(1) = 0$, because $2^0 = 1$.

On the interval $(0,1)$, we are looking at $\log _2(x)$ for $0<x<1$. Remember, $\log _2(x)$ is the number you raise $2$ to, to get $x$, but here $0<x<1$. Therefore, $2$ needs a negative exponent or $\log _2(x) < 0$. And, the smaller (closer to $0$) you want $x$, the bigger the negative exponent. There is a vertical asymptote.

If we switch the base from something greater than $1$, to something less than $1$, then all of the exponents flip. The graph flips.

Here is the graph of $y = L(x) = \log _{\tfrac {1}{2}}(x)$.

The intercept is $(1,0)$, because $\log _{\tfrac {1}{2}}(1) = 0$, because $\left (\frac {1}{2}\right )^0 = 1$.

On the interval $(0,1)$, we are looking at $\log _{\tfrac {1}{2}}(x)$ for $0<x<1$. Now we just need large positive exponents of $\frac {1}{2}$ to get small numbers. On the other hand, to get large positive numbers we need to raise $\frac {1}{2}$ to negative powers.

Logarithmic Functions

A Logarithmic Function is a function that can be represented by formulas of the form

\[ L(x) = A \log _r(B \, x + C) + D \]

where $A$, $B$, $C$, and $D$ are real numbers and $r > 0$.

The domain is all positive real numbers that make the inside positive.

Note: In the template for logarithmic functions, There is a leading coefficient for the function and there is a leading coefficient for the inside linear function.

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more examples can be found by following this link
More Examples of Elementary Functions

2025-01-07 00:50:25