Exponential and logarithmic equations and inequalities

Choose a common base

An exponential equation (equazione esponenziale) puts an unknown in an exponent. For a fixed positive base other than one, each different real exponent gives a different value. Consequently matching two powers with the same base is equivalent to matching their exponents. Try to recognise exact powers on both sides before turning to logarithms. Eight and one quarter can both be written as powers of two, for instance. A negative exponent changes a power into its reciprocal, so fractions often hide the most convenient common base.

For an inequality, first decide whether increasing the exponent increases the value. Repeated multiplication by a number above one increases a positive starting quantity. Multiplication by a number between zero and one decreases it. This order continues for real exponents. Thus with the same base above one the larger power has the larger exponent. With the same base below one, the larger power has the smaller exponent. When you rewrite a reciprocal base as a negative exponent of a base above one, use the direction rule for the new base, then handle the minus sign in the exponent as ordinary algebra.

Spot a repeated exponential

An expression with both $a^{2x}$ and $a^x$ contains a square and its base, since $a^{2x}=(a^x)^2$. Give $a^x$ a short name and solve the resulting quadratic. The new quantity is strictly positive for every real exponent. Keep that restriction next to the substitution and filter the roots of the quadratic before trying to recover the original unknown. A negative temporary root may be entirely valid for the polynomial while representing no possible exponential value.

A positive temporary root still needs translation back into an exponent. If the number is an obvious power, read the exponent directly. Otherwise the logarithm gives an exact answer: $\log_a u$ means the exponent needed on $a$ to obtain positive $u$. There is no need to estimate its decimal expansion to give a value or compare it with consecutive integers. Compare $u$ with nearby powers of the same base to bracket the exponent. Remember to reverse this comparison if the base is below one.

Restrict the logarithms before combining them

For a real logarithm (logaritmo), the argument must be strictly positive. In a sum of two logarithms there are two separate requirements. Their product can be positive when both original arguments are negative, so replacing the sum by the logarithm of a product can conceal forbidden inputs. List the original restrictions and take their intersection. You can then safely use the product rule on that domain. Subtraction of logarithms corresponds to division of arguments, with both original arguments again positive.

If two logarithms with the same valid base are equal, their positive arguments are equal. If they are compared by an inequality, their order follows the order of their arguments for a base above one and reverses for a base below one. Think of the logarithm as reading off an exponent from a power. For a decreasing exponential, a larger positive argument needs a smaller exponent, which explains the reversal without a separate rule to memorise.

Decide how much evaluation the question needs

A comparison with zero is particularly simple because every valid base to the power zero gives one. For a base above one a positive logarithm has argument greater than one. For a base below one a positive logarithm has argument between zero and one. Those endpoints perform different jobs: zero marks loss of the domain, while one gives a defined logarithm equal to zero. Give each endpoint its own test if the comparison allows equality. Finally, check any candidate in the original expression before reporting it. A substitution often reduces a complicated expression to an integer power you can verify in one line.