Graph PlottersPlot log(x) and transformed logarithmic functions while checking the valid domain and vertical asymptote.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
A logarithm is the inverse of an exponential function and is defined only when its input is positive.
HOW TO USE IT
The parent natural logarithm y = log(x) is defined only for positive x and grows slowly. This calculator uses y = A log(Bx + C) + D, where A controls vertical scale, B changes the input scale, C shifts the input, and D moves the graph vertically.
The input must satisfy Bx + C > 0. When B is nonzero, the vertical asymptote is x = −C/B. The logarithm has no horizontal asymptote and increases or decreases slowly depending on the signs of A and B.
Start with log(x), then try log(2x), log(x + 1), or a reflected and shifted curve. Use the graph cursor on the valid domain and zoom controls to keep the visible branch in view.
A logarithm is the inverse of an exponential function and is defined only when its input is positive. For y = A log(Bx + C) + D, the domain is determined by Bx + C > 0 and the vertical asymptote occurs where Bx + C = 0. The curve grows slowly, which is why logarithmic scales are useful for comparing quantities across a wide range. The signs of A and B determine whether the valid branch rises or falls.
For y = log(x + 2) − 1, the domain is x > −2, the vertical asymptote is x = −2, and the graph passes through (−1, −1) because log(1) = 0. Enter the example, stay on the valid side of the asymptote, and use the cursor to check the shifted reference point. If the input becomes zero or negative, there is no real logarithmic value to plot.
For y = log₂(x − 1) + 3, the domain is x > 1, the vertical asymptote is x = 1, and the reference point (2, 3) comes from log₂(1) = 0. The curve increases slowly as x moves right. Enter the expression, stay on the valid side of the asymptote, and use the cursor to test values such as x = 5, where y = 5 for this base-2 example.
The equation y = log_b(x) means b^y = x, so logarithmic and exponential graphs are inverses. Their graphs reflect across y = x when the same base is used. This relationship explains the logarithm’s domain x > 0 and why its growth is slow: multiplying the input produces an additive change in the output.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
The domain is x > 0 because the logarithm input must be positive.
For A log(Bx + C) + D, it is x = −C/B when B is nonzero.
Yes. Natural logarithm and eˣ undo each other on their valid domains.
The logarithm’s input must be positive. For log(Bx + C), require Bx + C > 0.
It occurs where the input equals zero. For log(Bx + C), solve Bx + C = 0.
A logarithm is the inverse of exponential growth, so large multiplicative changes in x become smaller additive changes in y.