Exponential and Logarithmic Concavity

Why Study These Functions Through Concavity?

Exponential and logarithmic functions are familiar from growth, decay, and inverse relationships. Their first derivatives describe whether they increase or decrease, but their second derivatives answer a different question: how does the slope itself change? For these function families, a short calculation often determines concavity across the entire domain.

The key is to track both the second-derivative sign and the domain. A function may have a second derivative that is always positive or always negative wherever it is defined. In that case, its concavity does not switch, so there is no inflection point within the domain.

Concavity and Inflection: The Sign Test

On an interval where a function is twice differentiable, \(f''(x)>0\) means the function is concave up, while \(f''(x)<0\) means it is concave down. A change in the sign of \(f''\) across a point is evidence of a change in concavity. The second-derivative sign test must be applied on intervals where the function is defined and twice differentiable.

For the exponential and logarithmic families considered here, the second derivative has a fixed sign throughout each domain whenever the function is nonconstant. This makes the domain especially important: a boundary excluded from the domain cannot be an inflection point of the function.

Exponential Functions

For a real base \(a>0\), the derivative rule is \(\frac{d}{dx}a^x=a^x\ln a\). Differentiating once more gives

Here \(\ln a\) is constant with respect to \(x\). Since \(a^x>0\), the second derivative is positive for \(a\ne 1\): the factor \((\ln a)^2\) is also positive. Thus \(a^x\) is concave up for every real \(x\), whether \(a\) is greater than one or between zero and one. The base affects whether the function increases or decreases, but not this concavity classification.

Coefficients in Exponential Forms

A common parameterized form is \(f(x)=A a^x+C\), where \(A\) and \(C\) are real constants and \(a>0\). Its derivatives are

If \(a\ne1\) and \(A>0\), the second derivative is positive everywhere, so the function is concave up. If \(A<0\), it is negative everywhere, so the function is concave down. The constant \(C\) does not appear in the second derivative. If \(A=0\) or \(a=1\), the function is constant and its second derivative is zero everywhere; there is no switch from concave up to concave down.

Logarithmic Functions

The natural logarithm has domain \(x>0\). Its first and second derivatives are

Because \(x^2>0\) whenever \(x>0\), the second derivative is negative throughout the domain. Therefore \(\ln x\) is concave down on \((0,\infty)\), and it has no inflection point in its domain.

For a logarithm with base \(b>0\), \(b\ne1\), the change-of-base identity is \(\log_b x=\frac{\ln x}{\ln b}\). The denominator is a nonzero constant, so

If \(b>1\), then \(\ln b>0\), making this second derivative negative. If \(0<b<1\), then \(\ln b<0\), making it positive. Thus a logarithm with base between zero and one is concave up, unlike the natural logarithm. In both cases, the sign remains fixed over the domain.

Parameterized Logarithmic Forms

For \(g(x)=A\ln x+C\), with domain \(x>0\), direct differentiation gives

When \(A>0\), the graph is concave down; when \(A<0\), it is concave up. When \(A=0\), the function is constant. More generally, for \(A\ln(mx+n)+C\), where \(m\ne0\), the domain is the interval on which \(mx+n>0\). Differentiating on that interval gives

The denominator is positive on the domain, and \(m^2>0\). Consequently, if \(A\ne0\), the second derivative has the fixed sign opposite to \(A\). The excluded boundary \(mx+n=0\) is not a point in the domain, so it cannot be an inflection point.

Why the Concavity Does Not Switch

The classification above follows from the structure of each second derivative. For example, \(A a^x(\ln a)^2\) is a constant \(A\) multiplied by positive factors whenever \(a>0\), \(a\ne1\). Its sign therefore cannot vary with \(x\). Likewise, \(-A/x^2\) has the sign of \(-A\) at every \(x>0\). The same reasoning applies to the logarithmic form with argument \(mx+n\): its denominator is squared, and it does not vanish anywhere in the domain.

In each nonconstant case, \(f''\) is strictly positive everywhere or strictly negative everywhere on the domain. Since there is no sign change, concavity does not change there. This is a complete reason that these standard forms have no inflection points; it is not enough merely to observe that their second derivatives do not equal zero at a few sample inputs.

A Common Sign Error

A frequent mistake is to differentiate \(\ln x\) correctly once, but then lose the negative sign in the second derivative. For instance, the incorrect calculation

treats the derivative of \(x^{-1}\) as if its exponent stayed positive. The power rule gives

The correction matters: it makes \(\ln x\) concave down, not concave up. Another check is that \(1/x\) decreases as \(x\) increases on \((0,\infty)\), consistent with its derivative being negative.

Summary

For exponential and logarithmic functions, compute \(f''\), identify the domain, and then determine whether its sign can change. The base of \(a^x\) does not alter the positive sign of its second derivative, while a coefficient can reverse that sign. For logarithms, the base and coefficient determine the sign; the natural logarithm itself is concave down. In these basic families, the sign remains fixed throughout the domain, so there is no inflection point.