American Journal of Business Management, Economics and Banking ISSN (E): 2832-8078 Volume 32, | January - 2025 P a g e | 31 www.americanjournal.org LOGARITHMIC FUNCTIONS, EQUATIONS AND INEQUALITIES Abdullayev Nurbek Shuhrat o'g'li Ro'ziqulov Sa'dulla Egamnazar o'g'li Tursunov Zohidjon Tohirjon ògli A B S T R A C T K E Y W O R D S This article provides information on logarithmic functions, equations, and solutions to inequalities. Logarithmic functions, equations, inequalities, graph, function, interval positive. Introduction Logarithmic function. let a > 0, a ≠ 1. Given that the number N is a logarithm on the basis a, the number A is said to be the degree indicator that needs to be raised to form the number N, and is denoted by logaN. By definition, ax = n (a > 0, a 1 1) is the X solution of the equation X = logaN number. The action of finding the logarithm of an expression is called logarithm of the same expression, while finding the same expression itself according to a given logarithm is called potentiation. when the expression x = logaN is potentiated, a recursive n = ax is formed. with a > 0, a ≠ 1, and N > 0, the equations ax = N and logaN = X are of equal strength. Thus we have a function y = logax (a > 0, a ≠ 1) that is continuous and monotonic in its field of detection. This function: a basis is called a logarithmic function. the Y = logax function is the inverse function to the Y = ax function. Its graph is generated by a symmetric substitution of the function graph y = ax with respect to the straight line y = X. Since the logarithmic function is an inverse function to the exponential function, its properties can be generated using the exponential function properties. In particular, the defining domain of the function f (x) = ax was D(f ) = {-∞< x<+∞}, and the domain of change was E(f ) = {0 < y<+∞}. Accordingly, for a logax function f(x) = D(f) = {0 < x<+∞}, E(f ) = {-∞ < y<+∞}. at a > 1, the logax function (0; +∞) is continuous in light, increasing, negative at 0 < x < 1, positive at x > 1, increasing from -∞ to+∞. Similarly at 0 < a < 1 the function is continuous at (0;+∞), decreasing from +∞ to 0, taking positive values at 0 < x < 1 range, and negative values at x > 1. The ordinate axis is a vertical asymptote for the logax function. American Journal of Business Management, Economics and Banking Volume 32 January - 2025 P a g e | 32 www.americanjournal.org Let's consider the following examples: 1.To solve 2x=4, we write 2x=22 and find the solution x=2. 2.Let 2x = 5. it is difficult to describe 5 on the right side in the form of a degree with a base of 2. But it is known to us that there is a real root of this equation. To solve such equations, the concept of logarithm is introduced. In general, the root of the equation ax=b (a>0, a≠1, b>0) is called the logarithm of the number b according to base A. Description: the logarithm of a number B according to base A is said to be the degree indicator that a number will need to be raised to form a number B, and is defined as logab. ax=equation B (since x=logab aloga B / B (1) < BR > can be written in the form. (1) the formula is called the basic logarithmic axiom, where a>0a≠1vab>0 2) Examples: 1) log2162) find the value of log50,04. 3) Solution: 1) since 16=24, it is necessary to raise the two to the fourth level to form 16, which means log216 = 4. 4) 2)it is known that 0,04% of the volume is 5%. Therefore log50, 04=-2 5) Examples: 3. we find log4 x to satisfy the equations 4) log x 4 to satisfy. 6) Solution: using the basic logarithmic axiom: 7) 8) xlogx 4 4, ya`ni x 34 4, x 4 34 1 we find the S. 9) 3 256 For any number a>0, b>0, a 1 1, b 1 1, x>0, y>0, and the real desired numbers n and m, the following equalities are satisfied: 1) log a 1 0, 2) log a a 1, 3) log a (xy) log a x log a y, x 4) log a log a x log a y, y 5) log a x n nlog a x, 1 6) log am x m log a x, 7) log am xn mn log a x, logb x , 8) log a x logb a 1 9) log a b , logb a These equalities arise from the properties of the exponential function. We will prove some of these. Using logarithmic axiom: x aloga x, y aloga y we find the. American Journal of Business Management, Economics and Banking Volume 32 January - 2025 P a g e | 33 www.americanjournal.org Whether or not we multiply these equalities by terms xy aloga x *aloga y aloga x loga y, x aloga x :aloga y aloga x loga y, is formed. y From these equalities come the Equalities 3) and 4) according to the definition of the logarithm. X to Alo X if we increase both sides of the mirror to n-level, X to Alo x yields and from this we find loga X to NLO. To prove that the formula for the transition from one-base logarithm to another-base logarithm is 8) in private 9), we proceed as follows: loga x gömək B gömək From both sides of the resulting expression x=ab we find a logarithm according to base b: b blogb a b logb x logb x logb a logb a Putting the value of b on the left side, we obtain the formula 8). If we say x=b from this formula, we get the formula 9). 5- example. If log25 a va log23 b bo`lsa, log23000 ni a va b express through? Solution: log 2 3000 log 2 (3 53 23) log 2 3 3log 2 5 3log 2 2 b 3a 3 6-xample. If log 3 x log 3 7 2log 3 5 3log 3 2 bo`lsa, x find. 2log 3 2 3 log 3 72 53 2 log 3 1758 , Solution: log 3 x log 3 7 log 3 5 From this x 21,875 Decimal and natural logarithms. Definition 1. The basis a=10 the logarithms that are decimal logarithms are called and lgx is expressed through, i.e. log10x=lgx 7- example.lg100=lg102=2 8: lg0,01=lg10-2=-2 2- description. A Natural logarithm is said to be a logarithm whose basis is a number e, and lnx is defined by, i.e. logex=lnx, e the number is an irrational number, e=2,7182818284… a in practice e≈2,7 can be taken as. Between decimal and natural logarithms lg x ln x 0,434294ln x and 1 ln x lg x 2,302551lg x there is a link. A in practice lg x 0,4ln x and lg e ln x 2,3lg x equalities can be used. 9- example. ln100, lge2 calculate. ln100 2,3 lg100 2,3 2 4,6. Solution: 2 lg e 2lg e 2 0,4ln e 0,8. American Journal of Business Management, Economics and Banking Volume 32 January - 2025 P a g e | 34 www.americanjournal.org References 1. Fundamentals of Algebra and analysis. Manual for academic lyceums (R.X.Vafoev, J.X.Khusanov et al. - T.: Teacher, 2003-368 P. 2.Fundamentals of Algebra and mathematical analysis. I k. Guide to academic lyceums (a.Abdukhamidov, A.Nasimov et al. - T.: Teacher, - 2007. 462 b. 3.Math. I, Part II. A manual for vocational colleges (a.Melikulov et al. - T.: 2003.