Question 268070
{{{y = log(5, (x))}}}<br>
I am not going to do all this for you. But I will help with:<ul><li>Finding base 5 logarithms, and</li><li>Determining the asymptotic line</li></ul>
Base 5 logarithms. Normally we would pick a value for x and use the equation to find y. But there is no calculator I know of that can find base 5 logarithms. So we have to find some way to convert the base 5 logarithm into a logarithm whose base your calculator knows (base 10 or base e (ln), usually). If you know the formula for converting bases of logarithms then you can use it to rewrite your equation in a form which uses only base 10 or base e logarithms. Personally I have trouble remembering the formula. So I figure it out each time I need it. Here's how:<ol><li>Rewrite the equation in exponential form.</li><li>Find the base 10 (or base e) logarithm of each side.</li><li>Solve this equation for y.</li></ol>
Let's see how this works:
1) Exponential form:
{{{5^y = x}}}
2) base 10 (or base e) logarithm of each side:
{{{log((5^y)) = log((x))}}}
3) Solve for y. We start by using the property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}} to move the exponent out in front:
{{{y*log((5)) = log((x))}}}
Divide both sides by {{{log(5))}}}:
{{{y = log((x))/log((5))}}}
So {{{log(5, (x)) = log((x))/log((5))}}} which is what the conversion formula, if we remember it, says that it would be.<br>
Now that we have the equation in terms of logarithms our calculator can find we are in a position to start "Evaluate the logarithmic equation for three values of x that are greater than 1, three values of x that are between 0 and 1, and at x=1."<br>
After you find the y values for each of the specified x values you have a bunch of ordered pairs to plot on a graph.<br>
For the asymptotic line, you need to find the domain of the equation. Logarithms must have positive arguments. So the domain of this equation is the solution to:
{{{x>0}}}. Then, if you replace the inequality symbol with an equals sign you get the asymptotic line. So your asymptotic line is x = 0 (aka the y-axis).<br>
Note the left side of inequality above is the argument of the logarithm, whatever it is. For an equation like
{{{y = log(a, (2x+1))}}}
The domain is the solution to
{{{2x+1 > 0}}}
Solving this we get:
{{{x > -1/2}}}
And the asymptotic line would be {{{x = -1/2}}}