The exponential function family
For any positive base b, f(x) = b^x is an exponential function. The most important bases are e (calculus, physics), 2 (computing, music intervals) and 10 (engineering, pH, decibels). All share the same shape. Different bases just stretch or compress the curve.
Graph properties
Every exponential function passes through (0, 1). For b > 1 it rises without bound and approaches the x-axis as x → -∞. For 0 < b < 1 the curve mirrors: it falls without bound on the left and approaches the x-axis on the right.
Domain and range
The domain of every exponential function is all real numbers, but the range is only positive numbers, since a positive base raised to any real power never produces zero or a negative result. As a result, exponential functions model quantities that can shrink toward zero but never go negative, such as radioactive mass or drug concentration in the blood, both of which approach but never reach exactly zero.
Inverse via logarithm
To solve b^x = y, take logarithm base b: x = log_b(y) = ln(y) / ln(b). This is how exponential equations are solved in closed form.
Common bases in practice
Base 2 shows up anywhere something doubles: computer memory, chess-board wheat problems, and doubling time calculations that ask how many periods it takes to reach a target multiple of 2. Base 10 is the language of orders of magnitude: the Richter scale, decibels and pH all move in powers of 10, so a pH of 4 is ten times more acidic than a pH of 5. Base e is the default in continuous models, the exact base used on the continuous growth calculator, because it is the only base whose exponential function is its own derivative.
From pure function to growth model
f(x) = b^x is the pure mathematical object; attach a starting amount and a time variable and it becomes a growth model. Write x(t) = x₀ × b^t and the abstract function turns into the exponential growth calculator's core formula, with b playing the role of the growth factor. The equation solver rearranges that same relationship to isolate any of the four pieces, initial value, base, time or final value, once the other three are known.
Exponential functions versus power functions
An exponential function keeps the base fixed and lets the exponent vary, as in b^x. A power function does the opposite: the exponent stays fixed and the base varies, as in x^b. The two look similar in notation but grow very differently. At x = 10, the exponential 2^x equals 2^10 = 1,024, while the power function x^2 equals only 100. Exponential functions eventually outgrow every power function, no matter how large a fixed exponent that power function uses, because repeated multiplication by the same factor compounds faster than any fixed-degree polynomial.
FAQ
What is an exponential function?
An exponential function has the form f(x) = b^x, where b is a fixed positive base and x is the variable exponent. When b > 1 the function grows, when 0 < b < 1 it decays toward zero, and when b = 1 it stays constant at f(x) = 1 for every x. For example, f(x) = 3^x gives f(4) = 81, showing how quickly even a modest base compounds as the exponent rises.
What is the natural exponential function?
The natural exponential function is e^x, where e ≈ 2.71828 is Euler's number, the unique base whose derivative equals itself. It appears throughout calculus, physics and continuous compounding because differentiating or integrating e^x never changes its form. At x = 2, e^2 ≈ 7.389, and at x = -1, e^-1 ≈ 0.368, so the same curve handles both growth and decay depending on the sign of x.
How do I solve b^x = y for x?
Take the logarithm base b of both sides: x = log_b(y) = ln(y) / ln(b). For 2^x = 50, x = ln(50) / ln(2) ≈ 5.644, which checks out since 2^5.644 ≈ 50. The same method works for any base; only the numbers inside the two logarithms change, so a natural log button alone can solve any exponential equation.
What is the difference between e^x and 10^x?
Both are exponential functions but built on different bases. e^x uses the natural base e ≈ 2.71828 and is standard in calculus, physics and continuous growth models. 10^x uses the decimal base and is standard in engineering scales like decibels and pH, where each whole-number step means a tenfold change. Converting between them uses 10^x = e^(x ln 10), since ln(10) ≈ 2.302585.
Why does b^0 = 1?
Any non-zero base raised to the zero power equals 1 by definition, not by coincidence. The exponent rule b^a × b^c = b^(a+c) only stays consistent if b^0 = 1, since b^a × b^0 must equal b^(a+0) = b^a. Setting b^0 to anything else would break every other exponent rule, so it is fixed at 1 for every base except 0 itself.
What is the graph of an exponential function?
For b > 1 the graph rises steeply, passes through the point (0, 1), and stays positive, approaching the x-axis as x moves toward negative infinity. For 0 < b < 1 the curve mirrors this: it falls steeply from the left, still passes through (0, 1), and approaches the x-axis as x moves toward positive infinity. Neither curve ever touches zero or crosses the axis.