About One-to-One Logarithmic Functions
A one to one logarithmic function pairs each allowed input with one unique output. It also pairs each output with one unique input. That idea matters when you need an inverse. Many algebra, data, finance, and science tasks depend on this property. The calculator on this page checks that condition while it evaluates a flexible logarithmic model.
Formula Used
The calculator uses the form f(x) = a log_b(c(x - h)) + k. The number b is the base. It must be positive. It cannot be 1. The value a changes vertical scale and reflection. The value c controls horizontal scale and reflection. The value h shifts the graph left or right. The value k shifts the graph up or down. These controls cover most transformed logarithmic functions used in class and applied work.
Domain and Range Rules
A valid logarithmic argument must be positive. Here, the argument is c(x - h). The domain is all x values that make this expression greater than zero. When c is positive, the domain is x greater than h. When c is negative, the domain is x less than h. The range is all real numbers when a is not zero. If a is zero, the output becomes constant. A constant function is not one to one.
Inverse Calculation
The inverse is found by isolating x. Start with y = a log_b(c(x - h)) + k. Subtract k. Divide by a. Rewrite the logarithmic statement as an exponential statement. The inverse becomes x = h + b^((y - k) / a) / c. This calculator uses that relation to solve a target output value.
How to Use This Calculator
Use the first group of fields to define the function. Enter base, a, c, h, and k. Then enter an x value for direct evaluation. Enter a target y value when you also want the inverse input. You can leave one of those two fields blank. Choose decimal precision for rounded answers. Press Calculate. The result appears above the form, so it is easy to review before editing values.
Reading the Result
The output includes the function, domain, range, monotonic direction, inverse formula, direct value, and inverse value. It also reports if the function is one to one. The steps show the main substitution, argument check, logarithm value, and inverse conversion. These details help you find mistakes in signs, bases, shifts, and domains.
Using the Examples
The example table gives quick test cases. Use it to compare common bases and transformations. A base greater than 1 behaves differently from a base between 0 and 1. A negative c value moves the domain to the left of h. A negative a value reflects the graph vertically. These examples make the behavior easier to see.
Exporting Your Work
After calculation, you can download a CSV file. You can also download a PDF summary. These exports are useful for notes, assignments, reports, and saved records. Always review the domain before trusting an answer. A logarithm is only defined when its argument is positive. If the input breaks that rule, the calculator explains the problem instead of forcing a false result.
Graph Check
This tool is also helpful for checking graph decisions before plotting. A one to one logarithmic curve passes the horizontal line test on its valid domain. That means no horizontal line cuts the curve twice. The monotonic report supports this check. It tells whether the function rises or falls across the full domain. This is especially useful when inverse notation appears in later solution steps.