This is true of the graph of all exponential functions of the form [latex]y=b^x[/latex] for [latex]x>1[/latex]. The curve approaches infinity zero as approaches infinity. We plot and connect these points to obtain the graph of the function [latex]y=log{_3}x[/latex] below. Describe the properties of graphs of logarithmic functions. This is known as exponential growth. All Rights Reserved. The most basic exponential function is a function of the form [latex]y=b^x[/latex] where [latex]b[/latex] is a positive number. This is because for [latex]x=1[/latex], the equation of the graph becomes [latex]y=log{_b}1[/latex]. Below are graphs of logarithmic functions with bases 2, [latex]e[/latex], and 10. At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function. With the semi-log scales, the functions have shapes that are skewed relative to the original. on the left, and increases very fast on the right. x Logarithmic Graphs: After [latex]x=1[/latex], where the graphs cross the [latex]x[/latex]-axis, [latex]\log_2(x)[/latex] in red is above [latex]\log_e(x)[/latex] in green, which is above [latex]\log_{10}(x)[/latex] in blue. Change the log to an exponential expression and find the inverse function. Thus, it becomes difficult to graph such functions on the standard axis. That is, the curve approaches infinity as [latex]x[/latex] approaches infinity. = Original figure by Julien Coyne. A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense. Taking the logarithm of each side of the equations yields: [latex]logj=log{(\sigma\tau ) }^4 [/latex]. -axis; replacing Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph. The primary difference between the logarithmic and linear scales is that, while the difference in value between linear points of equal distance remains constant (that is, if the space from [latex]0[/latex] to [latex]1[/latex] on the scale is [latex]1[/latex] cm on the page, the distance from [latex]1[/latex] to [latex]2[/latex], [latex]2[/latex] to [latex]3[/latex], etc., will be the same), the difference in value between points on a logarithmic scale will change exponentially. Graphs of [latex]log{_2}x[/latex] and [latex]log{_\frac{1}{2}}x[/latex] : The graphs of [latex]log_2 x[/latex] and [latex]log{_\frac{1}{2}}x[/latex] are symmetric over the x-axis. Under these conditions, if we let [latex]x=\frac{1}{b}[/latex], the equation becomes [latex]y=log\frac{1}{b}[/latex]. In fact, the point [latex](1,0)[/latex] will always be on the graph of a function of the form [latex]y=log{_b}x[/latex] where [latex]b>0[/latex]. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Replacing Before this point, the order is reversed. As [latex]x[/latex] takes on smaller and smaller values the curve gets closer and closer to the [latex]x[/latex] -axis. Then subtract 2 from both sides to get y – 2 = log 3 ( x – 1). The point [latex](0,1)[/latex] is always on the graph of an exponential function of the form [latex]y=b^x[/latex] because [latex]b[/latex] is positive and any positive number to the zero power yields [latex]1[/latex]. with Logarithmic scale: The graphs of functions [latex]f(x)=10^x,f(x)=x[/latex] and [latex]f(x)=\log x[/latex] on four different coordinate plots. of If the base, [latex]b[/latex], is less than [latex]1[/latex] (but greater than [latex]0[/latex]) the function decreases exponentially at a rate of [latex]b[/latex].