Definition: An exponential equation is a type of mathematical equation where a variable appears in the exponent (the power to which a number is raised). In simpler terms, it is an equation that involves an exponential function, which is a function of the form ( f(x) = a^x ), where ( a ) is a constant and ( x ) is the variable.
Basic Example: The equation ( 2^x = 8 ) is an exponential equation. Here, the variable ( x ) is in the exponent. To solve it, you can rewrite 8 as ( 2^3 ), so you get ( 2^x = 2^3 ). Therefore, ( x = 3 ).
In Real Life: An example in finance could be ( A = P(1 + r)^t ), where ( A ) is the amount of money accumulated after n years, ( P ) is the principal amount, ( r ) is the interest rate, and ( t ) is the time. This is an exponential equation that shows how money grows over time due to compound interest.
In more advanced mathematics, exponential equations can be solved using logarithms. For instance, if you have an equation like ( 5^x = 20 ), you can take the logarithm of both sides to solve for ( x ).
While "exponential equation" specifically refers to a mathematical concept, the term "exponential" in general can also mean "growing or increasing rapidly." For example, "The population is growing at an exponential rate."
There aren’t specific idioms or phrasal verbs related to "exponential equation," but you might hear phrases like "growing exponentially" used in everyday conversation to describe something that is increasing very quickly.
An exponential equation is a mathematical expression where a variable is in the exponent. It is useful in many areas, including finance and science, to model growth or change.