--- title: "The exponential function and the number e" description: "For a>0 and a≠1, y=a^x stays positive and multiplies by a for each unit increase in x. Its horizontal asymptote is y=0. For y=e^x, the tangent gradient equals the function value everywhere." canonical: https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ source: https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The exponential function and the number e For a>0 and a≠1, y=a^x stays positive and multiplies by a for each unit increase in x. Its horizontal asymptote is y=0. For y=e^x, the tangent gradient equals the function value everywhere. Subject: Mathematics · Area: Algebra · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/the-exponential-function-and-the-number-e-mt_2hyjN4JfJJ ## Ready when they can - Describe the shape, allowed inputs, positive outputs and asymptotes of y=2^x and y=2^(-x); calculate exponential values and identify fixed multipliers - Say what makes e different from any other base - Use a model of the form A e^(kt) to describe growth and decay ## Lesson: Growth that multiplies every step Compound change repeats a multiplier. For y=a^x, use a>0 and a≠1. Every real input x is allowed, and every output is positive. At x=0, y=1. Increasing x by 1 multiplies y by a. Divide the next output by the previous output to find this multiplier from values one unit apart. A base above 1 gives growth; a base between 0 and 1 gives decay. *(drawing: For y=3^x, each unit increase in x multiplies the output by 3.)* For y=2^x, the curve rises to the right and approaches zero to the left. For y=2^(-x), it falls to the right and approaches zero to the right. Replacing x by -x reflects in the y-axis: (x,y) becomes (-x,y). Negative powers give reciprocals: 2^(-2)=1/2^2=1/4. For every allowed base a>0, a≠1, the curve y=a^x stays above its horizontal asymptote y=0 and never reaches it. The tangent gradient is the curve's rate of change at one point. The derivative gives these gradients. For y=e^x, the gradient equals e^x everywhere. At x=0, both the value and gradient are 1. Here e≈2.718, so bases such as 3 are larger. An interval increase is the end output minus the start output; it need not equal the starting tangent gradient. *(drawing: This interval increase is e-1, about 1.718. The tangent gradient at x=0 is 1.)* For a positive amount, use N=A e^(kt), with A>0 and elapsed time t≥0. The exponent kt means k times t. At t=0, e^0=1, so N=A. Each time unit multiplies N by e^k. Positive k gives growth; negative k gives decay toward zero; k=0 leaves N unchanged. The rate of change is k times the current amount. This is a fixed proportional rate, unlike adding or removing a fixed amount each time unit. **Example.** Evaluate M=18e^(0.1t) at t=10. First calculate the exponent: 0.1×10=1. Then M=18e^1=18e. Using e≈2.718 gives M≈48.924. Calculate the power before multiplying by 18. **Recap.** For a>0 and a≠1, y=a^x stays positive. For y=e^x, the tangent gradient equals the function value. In A e^(kt), A>0 sets the start and k sets growth, decay or no change. *Lesson and practice text written for this dataset. Background sources studied: OpenStax Algebra and Trigonometry, sections 6.1 and 6.2, and Calculus Volume 1, section 6.8.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Compound Growth and Decay](https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf) ## Optional connections - [Recognising cubic, reciprocal and exponential graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9) ## Opens up - [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT) - [Hyperbolic Functions and Their Inverses](https://lightmysky.com/learn/mathematics/hyperbolic-functions-and-their-inverses-mt_gCUl1sWi9M) - [Euler's Formula and the Exponential Form](https://lightmysky.com/learn/mathematics/eulers-formula-and-the-exponential-form-mt_RX2IxsnSwG) - [Logarithms as the Inverse of Exponentials](https://lightmysky.com/learn/mathematics/logarithms-as-the-inverse-of-exponentials-mt_skLQLUan0T) - [The Boltzmann Factor and the Partition Function](https://lightmysky.com/learn/science/the-boltzmann-factor-and-the-partition-function-mt_u9MSoTgJkF)