Question

write the following equation as an equivalent exponential equation.
ln(10) = x

the equation ln(10) = x is equivalent to the exponential equation \boxed{}.

Explanation:

Step1: Recall the definition of natural logarithm

The natural logarithm \(\ln(y)\) is defined as the exponent to which the base \(e\) must be raised to obtain \(y\). In other words, if \(\ln(y)=x\), then by the definition of the natural logarithm, we can rewrite this in exponential form using the base \(e\). The relationship between the natural logarithmic function and the exponential function (with base \(e\)) is given by the formula: if \(\ln(a)=b\), then \(e^{b}=a\).

Step2: Apply the definition to the given equation

We are given the equation \(\ln(10)=x\). Comparing this with the general form \(\ln(y)=x\) (where \(y = 10\) and the exponent is \(x\)), using the definition of the natural logarithm (which has base \(e\)), we can convert the logarithmic equation to its equivalent exponential form.

From the definition \(\ln(y)=x\iff e^{x}=y\), substituting \(y = 10\) and the exponent \(x\) (from our given equation \(\ln(10)=x\)) into this equivalence, we get \(e^{x}=10\).

Answer:

\(e^{x}=10\)