## Key Equations

 General Form for the Transformation of the Parent Logarithmic Function $\text{ }f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$

## Key Concepts

• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for x.
• The graph of the parent function $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ has an x-intercept at $\left(1,0\right)$, domain $\left(0,\infty \right)$, range $\left(-\infty ,\infty \right)$, vertical asymptote = 0, and
• if > 1, the function is increasing.
• if 0 < < 1, the function is decreasing.
• The equation $f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)$ shifts the parent function $y={\mathrm{log}}_{b}\left(x\right)$ horizontally
• left c units if > 0.
• right c units if < 0.
• The equation $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d$ shifts the parent function $y={\mathrm{log}}_{b}\left(x\right)$ vertically
• up d units if > 0.
• down d units if < 0.
• For any constant > 0, the equation $f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)$
• stretches the parent function $y={\mathrm{log}}_{b}\left(x\right)$ vertically by a factor of a if |a| > 1.
• compresses the parent function $y={\mathrm{log}}_{b}\left(x\right)$ vertically by a factor of a if |a| < 1.
• When the parent function $y={\mathrm{log}}_{b}\left(x\right)$ is multiplied by –1, the result is a reflection about the x-axis. When the input is multiplied by –1, the result is a reflection about the y-axis.
• The equation $f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)$ represents a reflection of the parent function about the x-axis.
• The equation $f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)$ represents a reflection of the parent function about the y-axis.
• A graphing calculator may be used to approximate solutions to some logarithmic equations.
• All transformations of the logarithmic function can be summarized by the general equation $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$.
• Given an equation with the general form $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$, we can identify the vertical asymptote = –c for the transformation.
• Using the general equation $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$, we can write the equation of a logarithmic function given its graph.

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